Yes, your residence card (Aufenthaltstitel) allows you to travel within Germany, but does not allow you to cross the national border.
You are, however, required to show and/or hand out your passport and/or residence permit to an authorized official such as
police and customs officer
government employees, such as foreigners and registration officials
You need your passport https://www.gov.uk/uk-border-control/before-you-leave-for-the-uk As a US citizen, residence in another EU country doesn’t give you the right to enter the UK without your passport.
As far as travel rules are concerned, she may stay in another Schengen country for 90 days in any 180-day period. How does the Schengen 90/180 rule work? provides more detail but if she has spent 30 days elsewhere than Germany in January, she still has 60 days to use (at once or split in several shorter periods) between February and April so there is nothing ...
I'll give a purely practical answer without digging too much into the academics.
Your first concern needs to be getting on the flight. For this, airport staff will use the TIMATIC database, which states:
Passengers arriving from a non-Schengen Member State are not allowed to enter Germany.
This does not apply to passengers with long-term
right of residence (...
You are allowed to stay 3 months after losing your job, afterwards when your Blue Card may be withdrawn:
What happens if I lose my current job while an EU Blue Card holder?
If this happens, then you are allowed to stay unemployed for 3 months in order to find a new job otherwise your EU Blue Card may be withdrawn and you may be asked to leave the ...