Intrigued by this question I started a game of Iconians versus Terrans, Altarians & Krynn on a huge map, tough difficulty (usually I play on painful but as I'm not used to the Iconions nor to huge maps, toning it down seemed like a good idea)

I have to add that my advise will not lead to great scores or a fast game; my style is rather conventional and slow so there is plenty of room for improvement. However it worked at tough difficulty so I'm hoping this will help as a starting point...

I put all bonus points into economy and research, then chose industrialists as a political party (favouring the production bonuses over economy)

1) Build order:
Capital: 3 basic replicators, 1 precursor library, 1 precursor archive, 1 basic replicator, 1 dream conclave, 1 econ capital
Second & third planet: 2 basic replicators,  1 precursor library, 1 precursor archive, 1 basic replicator, 1 starport
Other planets: for now don't build anything except if you have 300% bonus tiles for production or research

2) Research order:
Precursor studies (get access to Precursor library for some more research capacity), Gravitational manipulation, Xeno engineering, Xeno communications all the way to Trade (Econ capital), Interstellar warfare until Space weapons (increase military production), New propulsion techs until Impulse drive (faster colony ships), Xeno biology & xeno medicine.
At that point it's time to improve your economy (here I had to increase taxes and settle for about 80% approval rating; until now I had 100% approval everywhere), so Interstellar bartering, Merchant emporiums, Merchant trade complexes.
You might want to increase taxes a bit earlier that I did, because here I had to sell some basic techs and influence points to minors in return for technologies that would give me passive bonuses + any cheap techs that gave me access to factories or labs. I also squeezed out at least 2000 bc from such sales to keep my economy afloat.
Until now I only built colony ships and the occasional constructor to claim resources. Now I let my capital build the maximum number of freighters I can now use and send these over to the Terrans and Altarians (they are further away so I expect my first clash to be with the Krynn)
I research up to small hulls and laser so I get access to the very basic defenders.
Research Interstellar refining, Molecular fabrication, Industrial replicator I / II / III (this took a long time, probably close to a year for the whole branch - you might want to stop at Interstellar refining and stick to Factories instead)

Around this time the Krynn were the only ones with any military force (about 2 dozen small & tiny hulls with up to 2 attack). They'd started colonizing lower quality planets in my territory and their boundaries were touching mine. Seeing that I have no military, they demand 1 planet. Without looking which one they ask for, I refuse, causing them to declare war on me. That's a bit earlier than expected... My 3 production planets start churning out defenders and within a matter of 2-3 months I have 1/4 of the Krynn's military strength. I have most planets defended by 1 defender, enough to fight off the weaker Krynn ships.

In parallel with the whole war thing I start filling any planets of 15 or more income with econ buildings, of course with the focus on social production. This was around July 2228 - by August I checked and saw that I had 14 colonies (most still not building anything)

I immediately research medium hulls and kinetic streams 1 & 2. That's enough to start building battle axes which can wipe out any of the Krynn fleets (up to 3 small hulls with 2-3 attack each).
Then it's time for planetary invasion and all the following techs. As soon as planetary invasion has been researched, I alternate battle axes and transports. A fleet of 1 battle axe + 1 transport is enough to invade a Krynn planet. I take a few with little population in my territory first. We both have 0 bonuses so my 1000 soldiers can easily invade any planets of less than 1000 population. As soon as I have access to mass drivers I attack just about any Krynn planet - even the ones of 8000 population fall to the onslaught.
I also research some of the basic miniaturization techs so my battle axes will have 10 attack from now on.

In the next months thanks to a few 1000 bc anomalies and a balance of between -50 and -100 bc per turn, I find myself with 3000 bc in cash. I upgrade my defenders to they have 4 attack; not a necessary move but it proved useful to keep fending off those pesky weak Krynn ships. Just make sure you have enough cash to keep paying for mass drivers during invasions.
I also spend some cash on spies as soon as my economy is making a small income (at about 60% approval rating, now it's more important to have enough cash to support my fleet and invasions rather than having some population increase bonus) and plant at least a few on each race.

Then I research the whole yellow branch with influence bonuses, until Cultural insurrection: I want to avoid losing any of the planets I just conquered to influence.

Pretty much continuing in the same mode, by July 2230 I have conquered a good dozen Krynn planets and they propose peace, saying this war is so expensive. Guess what, I turn them down and keep claiming more of their worlds. My military rating has been higher than theirs for a few months and keeps rising while theirs remains constant (my battle axes still wipe out their fleets with ease)
In the meantime I also formed an economic alliance with the Terrans - unfortunately I was too late to get a research alliance with either the Terrans or the Altarians, they kept refusing even if I threw in some cash.
By now my economic alliance, military strength (Terrans and Altarians still don't have any noteworthy military) and trade routes give me close relations with both and I ally.

I hope this helps as a starting point even if I didn't have the patience to go for an Immense galaxy or to complete the game (it's only a matter of a few months before I send over a few fleets to their capital and take it), don't hesitate to ask any questions...

So in the end it's quite a different approach, except for the one common factor of acquiring passive bonus techs and better econ buildings from other races.
Meaning that inherintly, the Iconians seem underpowered compared to most other races, though it's still possible to win through clever tech trading.

Best of luck!

Long trade routes enhanced with economic star bases also help.

Also being intrigued by the challenge, I took it up.  I had played Iconians before, but only twice, and that was some time ago and I couldn't really remember how they worked.  After playing them again, I am beginning to think they are not meant for a fast start, but are meant to claw their way from behind, which makes for a slow game. [Won by Tech victory at 7 years, 2 m, 4 wks at Tough.]

My settings were slightly different than the OP's.  Tough, Huge Map, Star Density =Tight Clusters Habitable Planets = Uncommon (Unc), Planets = Common,  # of Stars, Anolomalies, and Asteroids all = Abundant, Normal tech rate, 5 minor races, Frequent Random Events, and Frequent Extreme Planets (Hey, might as well exploit the Super Adapter ability!).  Full nine opponents (Left out the Korath and the Thalan since IIRC, they weren't in the original DL storyline). Mega Events off, No Tech brokering or Surrenders (if the AIs would surrender in a more sensible fashion I'd leave it on, but having a huge, but unarmed AI empire surrendering to an armed AI who can't even invade them irritates me.) Ascension and Diplomatic victories = off.

As the Iconians, I chose Technologists (my usual choice for any race, but with habitable planets at a premium, the +1 Sensor range seems more important than usual).  4pts went for +30% Economy, 2 for +15% Morale (I tax heavily), 1pt for Creativity, which was very useful in the early game since I got several early techs in rapid succession thanks to that. (Admittedly that was luck, because I've started another Iconian game, and despite the same race settings and Trading for the Arcean exclusive tech that gives +25% Creativity, it has not worked once so far despite a higher base chance of success!)

First Turn: Purchase Colony ship, purchase Precursor Archive, set taxes at 58% (63% approval). 

Tech path followed was similar to Noctilucus' in Reply#2, but changes after Xeno Engineering.  I went for Propulsion techs up to Impulse Drives and then Sensors to try to get early Survey ships out to keep the economy afloat by finding anomalies.  Xeno Comm up to Trade followed, then the Interstellar Bartering line up to Merchantilism (?).  Weapons techs followed, up to Kinetic Streams II, After that I started the Xtreme colonization line since I'd found some 18+ Heavy Grav Worlds.  Then I went for the long road to Merchant Trade Complexes.  Stopped recording after this, so can't tell you more tech details.

Fought three wars, the first against the Krynn thanks to the assassin RND event (we'd had good relations and trade until that point), the second against the Yor (for role-playing purposes - reclaim the homeworld!)  The third was against the Korx, who were the 400lbs gorilla of the game and who destroyed the Terrans.  Did not actually finish off the Korx, but stopped short after taking 70% of their worlds.  I was hoping to flip the rest, since the Iconians at the time were close to an Influence Victory, and I'd taken the high influence Korx worlds, leaving the weak ones surrounded by my influence.  Only got two planets this way, and I eventually went for the tech victory, letting the AIs fight among themselves ineffectually).

My current Iconian game has almost the same settings, but planets are now Unc, and the stars are scattered.  I have both starting saves (ToA version 2.03) if anyone would like to try them.  I'm not a strong GCII player so a more experienced player could probably get a faster win on that first game.  Are we allowed to post saves on the forum?

A question, why do the Iconians only get 7 customization points?  Is it because they start with a Class 14 HW? 

Edit 1: Yet the Thalans start with a Class 17 and get 8pts, go figure.

Two Iconian hypotheses that need further testing.

1. Early survey ships are important since anomalies vanish quickly, and are often worth more than tech trades.

2. Influence is important for the Iconians for passive tourist income and defense.  With inferior farm tech it is hard to fill transport and maintain the economy, so flipping worlds is more important than for other normal civs.