Can my local wallet be offline when the miner pool pays me to my t-add?

I checked my t-add wallet with zero balance, but mining pool indicates sending daily payout several hours before.

There are two quires here:

  1. How to check t-add after I created it few days before?
  2. How to update local wallet when my pc was offline before I check local wallet balance?

Yes. The payment goes on the public blockchain, and your wallet will sync with it when it comes back online.