Questions regarding wallets

I was thinking of verifying my wallet with Gemini or Uphold. But i am curious what exactly happens when my BAT is in one of the wallets?

I mean if the price of the BAT increases, will it be reflected in my wallets too?
Like if the price of BAT increase to 0.5 USD, will it increase in my wallet as well?

And also, is it possible to transfer my BAT to Brave wallet without using Gemini or uphold and simply in brave browser?

Fyi I am an android user.