I've not taken a Seiko apart, but I've done this to other watches.
Bezel really needs to come off to change the insert. Try it with the bezel in place, and you'll damage the old insert, the bezel or the crystal.
Pushers are retained by split rings (C clips).
If you want to give the case a good clean, think about buying a small ultrasonic cleaner. Used for cleaning jewellery, etc. Very cheap on eBay.
Bezel really needs to come off to change the insert. Try it with the bezel in place, and you'll damage the old insert, the bezel or the crystal.
Pushers are retained by split rings (C clips).
If you want to give the case a good clean, think about buying a small ultrasonic cleaner. Used for cleaning jewellery, etc. Very cheap on eBay.