1) Before the holes are patched, can I stuff the holes with insulation? If so, what would you recommend? I was at Home Depot some months ago and noticed these big rolls of pink insulation. They all have a different "R" rating. Is there any rating that is more suitable for my purposes? Also, should I get the ones with a paper backing?

big rolls are fiberglass batts. location determines R-value
requirement. paper backing is vapor barrier/paper to attach batts to stud faces. just fyi.

stuffing bits of insulation will replace what you've lost...
looks like rockwool or cellulose in walls. not sure which.
packing fg tightly reduces R-value, so don't compress the
batts. blown in insulation (like you have currently) has no
vapor barrier. in adding fg batts, just take paper backing off
the batt...then use just fg in holes.
you can buy lowest R-value..and just use more. the insulation is same R-value per inch...thicker is higher R-value..thinner (R-11-13) are lower R-value due to thickenss.

2) When the carpenter comes to patch the holes, can I use the same pink insulation?

not sure what you mean?

3) Is Home Depot's insulation good or is there a better alternative? Lowes? Another store? Online?

local to you.

that is more than 3 questions...the first question contains
3 questions.
just saying...one question always leads to more questions.

had knob & tube wiring?
once walls are repaired, air will not enter living space.
air comming through is part of leakage of wall to ambient..
and electrical penetrations. sealing holes in attic/basement
where wires were pulled through with caulk works well...but
electricians make holes...they don't seal them.
plumbers are the same way.
putting gaskets @ outlets & switches will keep help.

best of luck.