Homestay - intimate questions

I'm going to be in a homestay programm for 3 month in Tokyo.

I was wondering what was the norm for when girls have their menstruation.
At my house, we have to throw away the used sanitary pads in the garbage, but I had some friends who said they threw the in the toilets. What is the norm in Japan? I'm a little shy, so I don't feel really comfortable with asking this to my host parents when I arrive...
Also, which are the most common? Sanitary pads or tampons?
And is it okay to take baths in the same water as everybody while having menstruation, or considered dirty?

Another question, I shave my legs and armpits almost every days, I assume I can't do it in the bath, but is it okay in the part of the bathroom where we shower before entering the bath?

I hope someone will answer me, I'm getting a little nervous about these things!

we have to throw away the used sanitary pads in the garbage, but I had some friends who said they threw the in the toilets.
Never throw them into the toilets. But you'd have to ask your host mother exactly where you are supposed to put it; many households have this little trash box right in the toilet where you can place them, to be later put into the burnable garbage.

Sanitary pads or tampons?
Personal preference :)

And is it okay to take baths in the same water as everybody while having menstruation, or considered dirty?
If you use tampons, it would be OK to take bath in the same water. But if not, just wash/shower yourself outside the tub, and do not soak yourself in the water. Or, depending on what your host family is used to, you can ask to be the last person to take bath, soak in the tub water, then drain.

but is it okay in the part of the bathroom where we shower before entering the bath?
It should be fine there :) That's also where you wash.

You don't have to be nervous :) But you should not hesitate to ask your host mother/host sister about those things either.