No centralized crypto exchange will give you access to the private keys of your public addresses, they simply do not work that way.  What is important is that you have a wallet in which you can completely control private keys, fees when making a transaction and that you can always send coins from the exchange to that wallet.

No need to have another wallet, it would just mean higher costs for you in terms of fees you would have to pay to do the following - Coinbase -> third party wallet -> Nano X.

As for the fees when you withdraw from, say, Coinbase, you can't influence them - but when you send in the opposite direction (from Nano X) you can always check the mempool and see what the current fees are depending on whether you need a quick confirmation (1-2 blocks) or the time of the first confirmation does not matter to you. 

For me best site to check that info is : https://jochen-hoenicke.de/queue/#0,24h

All you need to do is look at the third graph, point you the mouse at the end of the graph and see how much satoshi per byte you need to pay to get in the next block which size is always around 1 MB - so anything below 0.99 MB gives a great chance for quick transaction confirmation.