Theses answers are going to depend on a couple of factors:
1) The particular exchange you are working with.
Each exchange is going to have its own set of policies and different levels of transparency communicating these policies to customers. Look for exchanges with more transparency/details.
2) The country the exchange assigns to you when you open the account.
Exchanges are facing financial regulations imposed by governments.
Global exchanges can have different guarantees in different countries.
For example, Coinbase and GDAX state, "If you are a United States resident, your Coinbase USD Wallet is covered by FDIC insurance, up to a maximum of $250,000." Source: How is Coinbase insured? This link Supported Countries shows the supported countries for GDAX and the supported fiat payment methods.
I made this video discussing insurance policies at Coinbase and GDAX. It may help: GDAX and Coinbase - Insuring your crypto is safe
3) If the exchange is centralized or decentralized.
If you are working with a decentralized exchange, you are in control of your funds. The Waves platform is an example of this: https://wavesplatform.com/
In general, it's not just the fiat that you have to worry about at centralized exchanges. If the government for example seized the assets of the company, they would also be able to control your crypto. If you are super worried about this, you can send your cryptos to a private wallet that you control. The downside to the private wallet is that you are totally responsible for security, backups, etc. There is no help if mistakes are made.
Most exchanges allow you to send crypto to them after you have opened your account. The crypto can then be used for trading. Additionally, many exchanges allow you to purchase using a credit card.
See this for details on exchange vetting:
How do I know which exchanges are serious actors?
I made this playlist that goes through policies of Coinbase and GDAX. It may help: GDAX Playlist