Gradual house rewire

" I will also run a new circuit for lighting for the house using 14/2. "

You will probably need more than one.

"My 1st question is, would it be possible to run new circuits with 12/2 and then branch out to each receptacle in a room using 14/2 from the junction box? Or would I have to use 12/2 cable throughout the new circuit?"

You could do that, but you would be limited to using a 15 amp breaker.

In general it is considered "bad form" because it might confuse people the future. But it is sometimes done on purpose where you have long runs to reduce voltage drops. But that would be more common in a LONG ranch house.

Probably not needed in an older 2-3 story house.

"And my 2nd question. After the rewire, will I have to remove the old wire inside the walls or can I just tape and cap them off (these wires will be no longer be powered)."

Yes, but the best way is to reach back in the wall and cut of as much as you can reach.

Then there is no question.

I might as well run all receptacle circuits with all 12/2 cable w/ a 20A breaker and run light circuits with 14/2 cable w/ a 15A breaker. Would this be over kill? Or will I be playing it safe.

Thanks again.

That is not uncommon to use 15 amp for lighting and 20 for receptacles.

Lighting often has more stuff that needs fishing and 3/ways with 3 conductors cables.

Much easier to fish 14/3 thena 12/3.