Ghusl

Bismihi ta'ala

If she did ghusl after her sexual climax, and has washed herself, then whatever the case may be after, she considers as pure. 

And Allah knows best. 

Bismihi ta'ala

As you would know, tayamumm is only done when one has no access to any water or using of water is harmful, like a sickness or extremely cold weather.

In any other case, especially if water is accessible, the mukallaf must do wudhu.

And Allah knows best.  

Bismihi ta'ala

You will not be able to do both wudhu' and ghusl simultaneously and at once. 

So, you can either do wudhu' before ghusl, or after ghusl. 

When you wipe your head during wudhu', you can use four fingers, or three fingers. Both cases are fine.

And Allah knows best.

Bismihi ta'ala

The discharge a woman experiences is tahir, even if it is due to lustful thoughts, as long as it was not an orgasm. 

She does not need to do ghusl, or anything else. 

And Allah knows best.

Bismihi ta'ala

If it is 'dirt', and it is accessible, and would prevent water from getting to your out layer of skin, then it is wajib to remove, for the purpose of wudhu and ghusl. 

If the dirt is not accessible, as in it is a stain under your nail that you cannot remove, or that the dirt is not an obstacle and will not prevent water from getting to your skin, then you do not have to remove it, and your wudhu/ghusl will be valid.

And Allah knows best. 

Bismihi ta'ala

Yes, ghusl or wudhu would be valid without rinsing mouth or nose. Such an act is not considered as wajib.

And Allah knows best.