Pity is seperate for each banner type … meaning if you roll on the weapon banner your pity does not increase or reset on the character banner (or the standard banner).
As for how pity works, every 90 rolls on the character banner is guaranteed to win you a 5 star. That 5 star has a 50/50 chance of being the special one shown on the banner. If you do not win the promotional character, the next pity roll (or normal pull of a 5 star) is guaranteed to be the promotional character.
Also … your pity remains even when banners change. So if you roll 80 times for Eula (and fail 80 times) … when the banner changes you only need 10 rolls to guarantee the pity.
Pity DOES NOT affect any other banner type though. So you have 3 separate pity amounts … one for the character, one for the weapon, and one for the standard banner.
As to winning a different 5 star … yes you can … assuming the 50/50 chance fails.
Hope this makes sense ?? … have fun.