Ha! ok, so, good questions, and the answer is a lot of "yes" and "maybes" :-)
FOR THE MOST PART, tho, no: support units are in separate battalions from the line battalions. You can see this in the Mech Handbook for the most part, and back in the old BattleForce breakouts (like in the 4SW products.) For smaller units, they would be on seperate lines in a battalion break out, but you almost NEVER (I can't think of any, but there might be a few) see the support units listed insde the company line chart.
(I'm assuming you mean stuff like logistical support, not vehicles and infantry, because yes, some units are combined arms down into the company level)
However, you will see that support teams (like technicians) are often "assigned" to mechs; that doesn't mean they're in that lance or company, just that they're always assigned there from their administrative hub. In the US Army, they might be listed as an "attachment/detachment," but for pay purposes, they're still listed under tech Platoon leader X.
This is a generalization, of course, for the Great Houses, most great powers, and most mercs: the Clans are TOTALLY different. :-)
(oh, finally, yes: if you have a support battalion, and companies A-C are designated to support the line battalions, the BN Headquarters company consists of two parts: the ACTUAL HQ platoon (BN CDR and his driver and stuff) and then the support platoons that ONLY support companies A-C: their maintenance, their food, pay and admin, etc. This sometimes confuses people: "why can't I task THOSE guys to help line battalion 1?" "because THEIR job is to support the SUPPORT companies, NOT to help YOU." :-)
hope this helps!
edit:
I will point out that BattleTech -usually- isn't so regimented whereas A-C support BNs 1-3 in that order, and you CANNOT EVER EVER break those up and shuffle them (like the US Army, sigh. Don't get me started)
BattleTech support units are generally more UK/Aussie in build, whereas they will have, say, an ammo company, a maint company, and a transport company, and then they will divvy up those assets as needed across the regiment. if BN 3 is in reserve, they might get more mechanics and less trucks than the BN 1 who is attacking.
Why this is important in Btech is, it keeps the handwavium going, and puts less "hard numbers" on things. Remember, the physics aren't real, the numbers aren't real, everything is best guesses by history majors, etc. hahaha
if you ever get into a spot where the numbers don't line up, pretend they do: that's battletech logistics.
