Ditto Bing on 1 and 2.
3.: "reducing" is what you'll see for shields when you are some nasty gas clouds. (You'll see "lightning" in the cloud when zoomed way out.) Generally, stay out of such clouds. There are no systems there (or very few), so you usually won't need to go there.
4. No. Fuel is stored at the planet itself. Small SBs are good, though, as a platform to put a hospital and rec center overhead, which makes the colonies happy --> reduced chance of revolts, and higher growth rate.
5. generally I don't build resupply ships, since there's usually fuel you can get at from planets, gas mines, or bases. And I put a fair fuel capacity on my ships.
As for stealth, it varies inversely with the ship size. So a resupply ship (being huge) gets very little benefit from stealth. I do not stealth mine. Also, resupply ships are only useful in a system, and I think that when ships are in a system they can be seen by any alien which has any ship in that system, regardless of stealth. (Anyone know different?)
6. I always use crash research only, at least after the very earliest part of the game. My chosen fields are:
All of those are good for getting speedy explorers and colonization ships, and troop carriers (which, along with salvaged ships of CA class and larger, are all you need for defense). Once I have gotten a good tech lead in the colonization techs, I start to mix in techs to allow myself to build decent battleships: torps, more shields, and of course shipyards. (Although very frequently if you explore aggressively you will get the large shipyard tech from a relic.)
- colonization tech to grab as much space as possible without conflict (and also more happiness from hospitals and rec, which means more tax income)
- if starting at low tech, the lower hyperjump techs (up to say Kaldos KH-7)
- some better engines than the lowest tech ones
- whatever is super-cheap, except labs