1. Is it normal for automated fleets to suddenly switch their home base to a gas mining station when I originally assigned them to a planet?
If you have Fleet Automation set in Empire Policies, then yes, it's normal for automated fleets to switch their home bases to random other points in your empire. If you don't have Fleet Automation set in Empire Policies, then this would be a bug. I personally would strongly recommend taking as much of the computer's fleet management authority away from it as possible; setting fleet formation and management to full manual and giving the fleet a home base and automation settings works and keeps the fleet where you told it to go.
2. Does the amount of stockpiling resources the freighters bring change when putting small/meduim/large spaceports on planets?
If I am not mistaken, the size and priority of a colony's resource stockpile is related to the size of the spaceport at the colony. However, I don't believe that this significantly affects the size of the cargo loads carried by freighters to the spaceport except possibly when there is a severe shortage at the location. Moreover, larger numbers of spaceports means more stockpiles that the freighters have to balance and distribute resources between, which means that some of your freighter time is going to be expended balancing the stockpiles rather than collecting new resources from mines.
3. When there isn't many resources around a planet and you want to really increase the population, is there anything else to do besides changing the tax down to 0%? I made a small space port far out for a military fixing situation but the build rate is so slow I need more population to increase it..
A medical center and a recreation center on an orbital base will help. You can also up the tax rate on some of your more populous worlds to make them less happy and more inclined to emigrate, and then hopefully at least some of them will immigrate to your underpopulated tax haven on your frontier.
A further suggestion, if you just need the spaceport, is to strip the station down to the bare minimum required set just to have something in the area (e.g. a command center + 1 shield generator + docking bay + construction yard + 1 missile + 1 energy collector + minimum required reactors + minimum required set of habitation and life support modules + 1 of each component fabricator + any other required components that I've forgotten - remember that if there isn't a red warning in the design screen, you can build the spaceport, and that yellow warnings can generally be taken as suggestions, and also remember that labs are not required). However, this will only help on future construction; if you want this to be done for the current station, you'll need to cancel the current construction project, which might not save you any time and will cost you some of the money you spent when you ordered the spaceport. If you don't require the shipyard capacity, you can use a stripped-down defensive base or a minimal starbase design instead.
And wonders that give + development for colony.
Wonders build too slowly on low-population worlds to be a worthwhile method of boosting the population growth. They're also much more likely to be completed by your competitors if you're building them on low-population worlds. A facility that boosts planetary development might be worthwhile, but I don't think I'd sacrifice a wonder for a chance to boost population growth.