I want to buy a hardware wallet either Ledger nano S or the Trezor

a couple of things im unsure about though, is it better/safer for me to transfer funds from exchange like bittrex to the hardware wallet then to another exchange or would it be better to go directly from one exchange to another?

Once the funds are on my hardware wallet I cant trade untill i put them back on the exchange? ( seems pretty obvious but just checking)

Can i transfer funds from my hardware wallet to someone elses exhange or to someone elses hardware wallet?

Is there any companys in the hundereds of cryptos that are trying to implament a direct hardware wallet to Fiat or even purchases dirctly from a hardware wallet?

thankyou