How exchanges generate transactions for all cryptocurrencies and send it to their network nodes? This is an important question for me, I know some Bitcoin wallets like blockchain.info are a Full Node, but exchanges which support so many coins, how do they create transactions per cryptocurrency? do they have a full node client for per cryptocurrency?

Just Bitcoin needs 170 GB space disk!

Are exchanges using third-party servers and apis?
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.
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.
SPV clients (like Electrum for Bitcoin) offer a middle ground between running a full node, and relying completely on a 3rd party service.
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.