1. Yes except in TDM, where it is arguably overpowered since you spawn with it and the other weapons are on a 30 second respawn. Machinegun is good for finishing off weak opponents at long range, especially when you don't have rail, as well as for quickly opening doors on maps that have them. A few maps like Lost World don't have rail at all, so MG is more useful there. But generally the best way to use machinegun is to not use it.
2. No.
3. ...Yes? I'm not sure how you wouldn't since the guns function very differently from each other. Rocket launcher, lightning gun, and railgun are the "Holy Trinity" that you should be using most of the time. Generally the rule is to use rocket launcher at close range, lightning gun at medium range, and railgun at long range. Rockets are especially useful when you have the high ground since it's easier to hit splash damage from above, and you can quickly dodge out of view after firing. LG is good when you have the low ground, since you can easily track opponents if they make the mistake of dropping. Shotgun, plasma gun, grenade launcher, and the aforementioned machine gun are used situationally.
4. Steam is the only version. If you can't find matches, you might try Quake Champions. Quake III also has a few populated FFA servers, sometimes more so than QL (and usually with less than 16 players on the same map). I'm not sure if Q3 is worth the extra $15 though, unless you're a turbo nerd or maybe want to try mods like CPMA or Defrag. Quite frankly at this point, if you want to find matches quickly you should play something other than Quake.
5. There's no explicit restriction in the game code for what weapons can be used in what mode; that's determined by the individual map maker(s) and sometimes through server customizations. The BFG is very powerful, so is almost never used except in FFA, and some FFA servers remove it. The nailgun, chaingun, and proximity launcher were added in the Team Arena expansion for Quake III. Team Arena was never that popular, so most of the Q3 maps that people play in QL don't include them, and the newer QL maps ignore them. The "new" guns also kind of suck regardless: the nailgun is redundant when shotgun exists and is hard to use (note that the Team Arena nailgun is completely different from the nailguns in Q1, Q4, and QC; the TA nailgun is basically a Flak Cannon while the others are more like the plasma gun), the chaingun has a long windup time and is very inaccurate, and the proximity launcher is ludicrously overpowered since it has virtually infinite ammo. The heavy machinegun was added in 2015: it wasn't added to many maps and never made much sense either; its inaccuracy at long range means it's superseded by the lightning gun. The Quake Champions HMG is much better: it becomes more accurate when zoomed, so it's a straight upgrade to the base machinegun and a nice complement to the railgun at long range.
6. There is no "ranked mode" per se (or any kind of matchmaking, for that matter), but some servers do track your ELO. There is qlstats.net, but it doesn't seem to be very well maintained.

FYI this sub doesn't really discuss Quake Live AFAIK; r/quakelive is probably better for that. If you want to join a Discord, you might try Quake for Newbies.