Prayer

There’s no difference between a man’s and a woman’s prayer.

No problem if they’re clean. If you’re a man then you have to change and then go to the masjid.

Whenever you enter the masjid, you must join the Imam regardless of the position he is in. If you manage to perform ruku while he is in ruku, this Rakah is valid.

This depends when Fajr is due. You must not depend on the Hanafi timetable as their Fajr and Asr are wrong and not according to the sunnah.

Yes.

You don’t have to offer your sujud on a cushion, but you must make your sujud by bending more than ruku’ while placing your hands on your knees.

The man is not sinful as he is not obliged to wait for you, and your salah is valid.

You must wait for the two salams.

The prayer is valid.

If the Time was truely due, one can pray even if the athan wasn’t called. In Hanafi masjids, they delay Fajr and Asr prayers. One can pray before them as per the majority of Muslims.