I do not believe you can roll one 401K into another. You'll have to roll it into an appropriate IRA. I would also recommend rolling company A 401K into an IRA... Best to cut ties with your old company's 401K program and take complete control in an IRA.
That's not correct; you can roll an old 401(k) into your current company's 401(k).
OP, if you do backdoor Roths, you're right that you should not be rolling anything into a traditional IRA. I would do the same thing you are doing: roll at least company B into company C (and I would consider rolling over company A as well for simplicity). Typically, the main reason not to roll over an old 401(k) is if it has better terms and investment options than your new 401(k). If your new 401(k) is just as good, the only reason not to do a rollover AFAIK would be the hassle (also, sometimes your funds will be out of the market for a few weeks while the rollover is in process).
Personally, I have an old 401(k) I'm not moving because (1) I also do backdoor Roths, so won't roll into an IRA, and (2) my old 401(k) has slightly better/cheaper investment options than my current one (namely, access to VINIX). My current 401(k) also has pretty good and inexpensive investment options so it wouldn't be crazy to consolidate them for the simplicity (and ability to completely cut ties with my former employer), but to date I haven't concluded that it's worth the hassle.