I've always used a "non-native" wallet to mine to, or else Nicehash wallet. I am just now starting to use a wallet where the blockchain is stored on my computer, while mining with an ASIC. Do I have to keep my computer on and the wallet software open in order to mine to this wallet address? I am specifically mining Pirate (ARRR) to an Agama wallet on Windows, using an Antminer Z11, and mining with Piratepool.io.
You are mining at a pool. This means that you will receive payments now and then. Great.
Now, you have to learn something important about cryptocurrencies: they are
not stored in the wallets. The blockchain holds all the information about the transactions, thus who has how much money.
That's why for many coins, including Bitcoin, if I give you a wallet address, you can send money there and you can see on a blockchain explorer how much money is there.
The wallet "only" handles the private keys, the "things" that allow you spend those coins.
That's why
you don't have to keep your wallet open to receive payments, that's why you can have the same wallet on multiple computers, that's why if somebody steals the wallet info (private keys or seed) from you can steal your coins.