Will Mom still be able to claim her home as her primary residence?

I believe a primary residence is where one lives more than 6 months out of the year.

If your Mom plans to sell her house sometime in the future, to qualify for the tax exclusion of $250k, your Mom would need to have lived 2 years out the last 5 years in that house. If she misses it by one day, the house becomes an investment property, thus no tax exclusion.

Do you mean for voting purposes? To maintain the homestead exemption?

I would think she would consider her new address as her primary home, but I guess I'd need an explanation of what you mean.

Who will be living in her house when she leaves it?

Claim it for what? Not sure what you mean.
For us, when mom went into ALF, we rented out her house, so the city said she could no longer have the senior or veterans tax exemption. The rent money helps pay for the ALF.