Once smart contracts are live with Cardano and fee's are being paid is the structure similar?
There are already fees today that have to be paid with every transaction. But there will probably be more transactions and therefore more fees when smart contracts are enabled with the Alonzo hard fork.
That being said, a stake-pool in Cardano works differently than a Bitcoin stake-pool. A Bitcoin stake pool usually lets miners connect to the pool and then everybody tries to find the right nonce for the same block. All the rewards go to the owners of the pool because they own the Bitcoin node that will submit the block when one of the miners has found it.
The proof of stake mechanism on Cardano doesn't require miners. A stake pool is just a single node that produces blocks when it is assigned a block by the network. No race for beeing the fastest to brute force some random number. As an ADA holder you can choose to delegate your ADA to one of the pools. The more ADA is delegated to one pool, the higher the chance that this pool will get a block assigned.
When a Cardano stake pool produces a block, it gets rewards. The rewards consist of the "remaining ADA" and the transaction fees of the transactions in this block. The rewards for this block are distributed among the wallets that are staking with this pool automatically. The pool owner can decide how big of a cut (fixed cost + margin fee) they will keep for themselves. The rest of the rewards are distributed among the delegators according to the size of their stake. This is all done as part of the Cardano protocol and the stake pool operator doesn't have control over the rewards of their delegators.
Will staking pools have their own transaction fee payout structure?
Some pools take 100% of the rewards. Those are called private pools. Binance runs pools like that for the people who want to stake directly from their exchange. The rewards then get distributed by Binance themselves to the accounts of their customers.
There are projects that do something called an ISO (initial stake pool offering) where they take 100% of the rewards and in return pay their delegators in some other token.
And if so will the busier the network equate to higher the staking rewards?
Yes. The more transaction fees a block contains, the higher the rewards.