The same kind of business trust and legal things that let an exchange or a bank running I suppose.
There have been exchanges doing what you said in the past, and they have gotten the same legal pursuit as any other business scamming his clients.
Here is some links: https://www.coindesk.com/six-arrested-over-cloned-crypto-exchange-that-stole-e24-million
Sounds that right way for you would be to run Bitcoin Core (full node) and develop your own deposit / withdrawal system from scratch. Relying on third party APIs has potential problems with reliablity, security and user's privacy.
If you plan that single user can deposit BTC multiple times, best practice is to change deposit address with each ...
https://blockchair.com/ is pretty good for data analysis on 9 different blockchain it allow you to filter by lots of settings like date, reward, hash, output etc... You can also directly download the data they offer in different format.
And they also have an API: https://github.com/Blockchair/Blockchair.Support/blob/master/API_DOCUMENTATION_EN.md
I want to transfer coins to multiple exchanges, how can I do it?
Just open accounts on all those exchanges and then get deposit addresses. Transfer funds howefver you like.
Can I split coins from one wallet into the exchanges?
Do I need a specific kind of wallet?
You can just withdraw from wherever your coins are now to your deposit addresses ...
Any service or exchange allowing conversion of one asset into another (whether it is decentralized or centralized), is eventually executing the trade on an orderbook with a large exchange or liquidity provider.
Larger exchanges like Coinbase, Binance, Bitstamp etc. operate their own orderbooks, which have liquidity from users and market makers.
Second tier ...