I'll confirm this. I've had people go nuts with mini-launchers before, and I think I've only seen relation deteriorate with only one group, and I'm not too sure if it was from the launcher-happy.
As for groups...
Government: Important, at least in the early game. They pay you weekly, but never more than 50% of their own budget. Your perfomance each week can give you an increase(or decrease) in funding. Do badly enough(intentionally), or they become hostile for any reason(alien takeover, you attack them), they'll terminate funding, which is permanent. Late in the game, you can usually fund yourself selling self-produced or captured alien goodies.
Marsec: Supplies most of your heavy hardware and vehicles, so never, ever irritate them or let them be taken over by aliens.
Megapol: Another major equipment supplier, a hostile Megapol has the added Fun of police cars attacking you.
Cult of Sirius: Always hostile, but there's bug that causes them to demand reparations to go to Unfriendly. It quickly detiorates to hostile. Early-game not much of a threat, late game the aliens supply them with heavy gear. Their vehicles will attack yours, but they're never a threat.
Superdynamics: Moderately important, they supply (air)vehicle engines and fusion powerfuel.
General Metro: Supplies lighter ground vehicles and their engines.
Cyberweb: Supplies targeting computers for vehicles(which are next to useless) and nothing else.
Nanotech: Supplies medkits and nothing else.
Sensovision: Worthless*. Roleplay-wise, it's a form of mass-media, but really, they're not that important.
Gravball League: Again, worthless. Those gravball stadiums are nastly places for a firefight, though.
Sancuary Clinic: Again, worthless. RP-wise, they control hospitals, but that hardly matters for you. You (should) have at least one medical bay at your base.
Transtellar: Important, without them, you can't transport items between bases.
Solmine: Supplies Elerium-115, which is ammunition for plasma-based vehical weapons.
Lifetree: Worthless. RP-wise, they run the schools.
Nutrivend: Worthless.
Evonet: Worthless again.
Energen: Some more worthless.
Synthmesh: Presumably worthless, although according to the UFOpaedia, they repair damaged buildings.
Psyke: Only significant because they'll attack you if hostile(they also fight police constantly). Raiding them might yield more psiclone, which can be sold.
Diablo: Supplies incendiary grenades, but otherwise functions like the other gangs. Probably more dangerous than the other two gangs, because of those grenades.
Osiron: Another gang, probably less dangerous than Diablo, probably more dangerous than Psyke.
S.E.L.F.: If they become hostile, you can no longer hire androids.
Mutant Alliance: If they become hostile, you can no longer hire mutants Sectoid Hybrids. (darn political correctness)
Extropians: Useless political party.
Technocrats: Another useless political party.
*It's possible that corporate entities, probably the political parties as well, are taxed by the government faction, so if you wreck the corporations' balance sheets, the government will have less money, possibly crunching your funding. Then again, bombing the corporations might hurt your rating anyway.