Re: Cloth wrapped BX cable replacement
If it is brittle you should replace it when you touch it. As for the grounding, you cannot use the armor plating as a ground for any new work, also remember if you take out a section of the old cable, you will ten be taking out the ground for that section.
My guess is the electricians will recommend if you are replacing just an outlet you install a GFCI receptacle and use the "no equipment ground" label.
You may want to invest in a residential code book as you are tackling a lot of work. They can be found for less than $20 at on.line and b&m retailers.
Good luck, and post pics, before and afters are great(at least I think so).
It's this old house, not this built after your dad was born house.