If I have a segwit address and I want to send BTC to the exchange, which supports only legacy addresses, can I even tranfer the money?
'Having a segwit address' dictates how you prove you received/own the money, not how you spend it. If you have a segwit UTXO, it's perfectly fine to create a transaction with witness inputs and then normal old P2PKH outputs, sending to 'legacy addresses'.