بسم الله الرحمن الرحيم
(1) Yes, you may do so. But it is not good to take bath naked.
(2) Yes, it is correct.
(3) If your satr is covered also the portion of bathroom is separate from that of toilet and there is no bad smell etc then you may read Bismillah etc slowly there. There is no need to recite the verses of the Quran.
(4) You may take bath in both the ways, there is no harm in it according to Shariah.
Allah knows Best!
Darul Uloom Deoband