Well, holding an 85% approval tells me your taxes are too low. No, seriously. Due to the way the game works, you only need >50% approval to A) grow pop (actually 41% or better at 1x rate, 75% or better at 1.25x rate, and 100% at 2x rate) or the single week that an election takes place.

I threw an idea out there of having mini-elections each week, but thus far no one has even commented on it. As it stands, the system can easily be abused in this way, and I don't know anyone who doesn't-though they may be doing so to different extents.

In any case, while you can technically dip as low as 21% after your growth is done (per planet basis, so overall may be higher than this, as some planets will more than likely have a lower approval due to pop/quality/etc), you can certainly go as low as 41% (not 40%, mind you, as then your growth shuts off-this is particularly important when you're cranking out transports...) or ~50% approval with no real downsides.

The above is planetary approval, however-your overall approval will probably be higher, as that's going to be the minimum for any given planet. You will probably want to crank up your overall approval a couple of notches the week or two before an election, particularly until you get the hang of the timing. I've found that in most cases, an overall approval of ~60% is sufficient to not lose any senate votings. 50% is iffy, as sometimes you lose, and sometimes you don't. For me, if I'm using a political party, I need those bonuses, so I'd rather keep them by losing some cash that turn.

The other thing you may want to consider is population. Based on your 85% approval, I'm guessing you don't have a population problem-which is a good thing. In general, you don't ever want more than one farm on a planet, and you also in general never want one on any civilization capital (i.e. your homeworld). However, you DO want that single farm, when you can get it, and after pop has reached the initial colony's 6B cap.

I hope someone else can answer your influence questions-to be honest I'm still a little fuzzy on that myself. What I can gather is that you A) need a minimum of 4x the native influence on the planet (viewable when you've reached either low or medium spying, I forget which) and any value above 4x doesn't really matter. As far as anyone can tell, a planet with 100x the native influence on it won't flip any faster than a planet with only 4x. Though, you may want to check if that particular planet that refused to flip had a Re-Education center on it...that makes it immune to cultural flips.

Which reminds me of the game I had a planet flip the turn it built it, but that's another story.

As a note on the economy: Ship maintenance becomes a killer late game-and, in TA, all throughout. That's probably your biggest culprit, but I doubt it's the only one.

Reading between the lines, I'm guessing that you did not build a farm on each planet to increase population. Population impacts tax, tourism and trade revenue; it also impacts your influence. A farm, entertainment building and economy building on each planet should have you rolling in cash by mid game. I build a second entertainment building if population will exceed 16b. Try not to go over 20b or you will have approval problems. Build more economy buildings on high population planets because they add a substantial bonus to your tax revenue - 30% for stock market.

Stock market will also amplify influence (5%) or you can build cultural exchange centers for a 25% bonus. Much cheaper than a starbase if you can spare the tiles.

Altarian is a good choice for influence victory because the AI will preferentially surrender to you (assuming you choose good alignment). That will make it a lot easier to bring 75% of the map under your influence.

Ships cost money to build and maintain. Planetary improvements cost to build, maintain and use. After you invade the planets population will be your surviving soldiers - who probably aren't paying enough taxes to cover the cost of operating the colony. If you flip, you get the entire population and the colony will probably be acretive to income immediately.

If you want a peaceful victory, maintain a powerful military. I play neutral and find that I can usually keep the AI off my back as long as my military rating is above 100. If you are playing good, you may need more than that to keep the evil civs from attacking you.

The one thing I wish to toss out there is that perhaps you didn't have enough economic buildings down to offset the costs of your labs/factories. Oftentimes a planet with 1 farm, 2 morale buildings and a ton of econ buildings will help offset many other lab/factory planets.

If you see that you have too much spending on research/military/social/ship maintenance/colony maintenance then you may wish to remove some of your labs/factories and replace them with a market. Heck, 1 market per planet may help stem the tide if you don't want an entire planet devoted to your economy.

Your espionage level must be at Low (the first level you reach after placing spies) in order to see the IP ratio. Once you hit 4.0 on a planet, it has a chance to flip.

To push things along faster, place an influence base in the same sector and near the planet. Influence bases don't really work as well in the adjacent sectors like planetary influence does. Get one or two of those near by one of their planets and keep adding influence modules to it (the ones that will take you to 100%, IMHO don't bother with the second line of influence modules). If one base with the 100% module in place doesn't take them to 4.00, then build another base. Just a note, they will NOT like these things around and may get lippy with you for placing influence bases near them.

Oh yeah...and don't be afraid to shut down a farm if they have one to reduce their population cap. As others have said, IP is related to population. If you can reduce their population with a spy, then you can possibly get to the 4.00 ratio you need.

New player myself, but can think of one other reason why it may not have flipped. That race might have had the loyalty trait (Yor get 100 loyalty, others get some less, and it might be taken as a random bonus) which makes it harder if not impossible to flip them culturally.