No, unfortunately not.

Perhaps, I think January 1 is open. The pool zone is mixed with men and women, but the baths are separate.

Yes. When you pay entrance fee (2310yen), you can borrow a towel, a bath towel and hall clothes.And you can enjoy several areas. For example,indoor&outside spa area(mixed gender zone ,wear swimsuits),hot spring area(gender-segregated zone),garden area and rest area. If you go after 6 pm,the admission becomes cheaper. (1295yen)

can enter

Yes

Yes!

一駅離れますが、練馬の文化センターの所に、そこそこ大きな駐車場があります。

Bade pool is a mixed bath meaning that it is a swimsuit.