Proof of stake coins can't be mined. Quite a weird question.
Actually there's at least one coin I know of, which used to have some hybrid PoS/PoW. I didn't read their thread much lately, so I don't know if that has change, however, you can take a look there.
It's
Magi Coin XMG.
Can you offer staking rewards if the token cant be mineable and the project doesnt have its own wallet?
If the coin is 100% premined, no further mining is necessary. You can buy the coin. (Of course then mining rewards cannot exist).
One can offer you some sort of "depositing" rewards which may or may not be related to staking. It can be staking, see LPoS/Waves, staking by leasing; but some others may just use the term "staking" for benefits/rewards.
A wallet only handles private keys. The coins are still on the chain, with or without wallet. Some sort of wallet is required for the users be able to use the coin. But I am not sure what you mean by "own" wallet. Some coins have developed websites for doing this job, it's still own wallet, but it's less secure imho and not so intuitive to be "own".