1. For an exchange, 170GB is not a lot, at all. In fact, it's a laughably small amount. Even on an SSD (which is not necessary), the storage cost would be about $100. Big exchanges might make $1M in revenue every day. Operating a node (or ten) is a complete non-issue.

2. You seem to be thinking that if Bitcoin's blockchain is 170GB, then so are the others. But Bitcoin is the most popular cryptocurrency, and it has way more transactions than any other (with the possible exception of Ethereum, which is more wasteful and has high traffic due to the ICO hype). The blockchains for most cryptocurrencies are tiny in comparison, and the sum total storage shouldn't be more than, say, 1TB (again, tiny for any large business). A single machine could run nodes for all coins.

3. SPV clients (like Electrum for Bitcoin) offer a middle ground between running a full node, and relying completely on a 3rd party service.

4. Ultimately, every exchange decides how to set up its infrastructure. It might run a node. It might not. There's no real way to answer this in general, this is something which should be asked of each specific exchange.

Funding is the answer to your unasked question. A standard 10U blade server can hold 24 servers--you can fit 4 in a standard rack and still have room for a 150TB NAS and routing. A modest setup could be had for under 100k and could easily support full nodes of a hundred cryptocurrencies.

No reputable exchange would host their wallets, private keys, and critical customer data on 3rd party servers. There's just too much at stake.