1 - Yes.  Only ships equipped with a survey module can enter a wormhole.  Putting a survey module on a colony ship, especially early in the game won't do you much good.  You still need to have the range to colonize a planet.  If you don't, you'll just cruise past any habitable planets until you're within your range.

One useful trick could be to outfit a constructor with a survey module.  That way, you can build a starbase (of any type) and extend your range in that direction once you're through the wormhole.  It's a risk though.  You might wind up so far away that building a starbase takes way too long to be useful or you might pop up right next to your home planet. 

It's better to just accept the game as a game and punch through by other means.

2 - Again, yes.  Research points go toward you empire's research ability.  Individual planets don't research any specific thing.  You choose where to allocate your abilities through the research screen.  However, a planetary bonus or starbase assist to research can boost a single planet's research output.  A galactic achievement can boost your overall ability.

3 - Economic resources add to your overall economy.  I'm not sure what the exact calculation is.  They can be handy as a little bonus, but I wouldn't chase them down.

In fact, any resource can be handy (except under very specific strategies, the influence resource can be a liability).  My favorites were Military (red) and Morale (yellow).  Morale can help boost your population, boosting both economy and influence.

MottiKhan is right, but I might add that you can enter a wormhole with a fleet of ships, and in that instance only one ship of that fleet needs to be equipped with this Survey Module. The other can be whatever you like.

You can, of course, equip every ship you want with such a Survey Module, but as it is expensive it would be a waste to stuck it on Constructors/Colonizers (as these ships break down after constructing/colonizing) or Fighters (who cost maint). Fast long-range vessels build solely for surveying are best. You can add them to any fleet with which you want to enter such a wormhole, and release them after the jump to go for other anomalies.

In the earlier stages when playing against the Dread Lords this is actually a good way to evade them successfully, even when they match the speed of your ships.

The idea you described above is a pretty good one, I've never did that but it might work out. Stash a Surveyor with a Constructor and the rest all Colonizers - make the jump, built a base for reach and then go colonize.

To be honest, it's phantastic idea. Imagine an Immense galaxy with all the AI's having one area, which usually is more like a circle, a coherent mass. Using the above method could result in having to manage/built up 2 or 3 etc independant areals, as they might be out of reach. It might be an advantage, but also an increase in difficulty, as other races would be more willing to declare war on you. But even if one superior neighbour goes hostile and conquers one such areal, you still could be playing in the other remaining one, which probably are safe because out of reach.

I need to keep that in mind for my next game.

No, because the game lets you build a base out of range.