Is there any potential risk of losing all my ADA coins when staking with Exodus partner
As you can see on the everstake website they have a pool called EVE1. This is a normal Cardano stake pool. You can see some details about this pool on adapools.org (or any other site that provides a similar service).
The pool has a 3% fee, this means you can stake with it from any Cardano wallet without risking your ADA. One thing that is very important: The ADA will always stay in your wallet, you never have to send it somewhere else. If some stake pool requests you to send all your ADA to another address, it's a scam.
Generally it's recommended to use one of the official wallets for storing your ADA. This is either Daedalus or Yoroi. But as long as you control your keys, also other wallets can be used to stake ADA as long as you trust the creators of the wallets.