You work out of your own depot only and sign the routes and traction for your depot. The two different brands don’t cross over.
There could well be exceptions to each of these I guess, but I’ve not heard of any.
I believe the West Midlands depots on the Wolves - New Street - Coventry corridor cover both West Midlands and LNWR work at the moment, though it is a franchise commitment to separate them out at some point. I certainly see both Green and orange ties at Liverpool Lime St on LNWR trains for example and some of those form mixed services, like the Crewe to Walsall service.
Correct. Birmingham - Liverpool services are worked predominantly by Crewe depot (LNWR) but also Wolves and New Street (both WMT). Similarly Birmingham - London services north of Northampton are worked by New Street and Coventry (WMT) but also by Bletchley and Northampton (LNWR). There are a few other examples as well. As LowLevel says, the giveaway is the ties!
If the depots really do get split as planned it's going to lead to a much less efficient railway.
Getting back to the OPs original question, it's worth pointing that at many TOCs, including this one, new drivers often start off with only signing a limited amount of routes and/or traction. Sometimes this is done through links (for example, AFAIK the junior most link at New Street only works a few routes closer to Birmingham - they don't go to the likes of Liverpool or Shrewsbury) and sometimes it's just a case of only working one or two routes for a few months before learning other routes.