For the title question, it depends on the person in question and how heavy the flow is that day, etc.

Haven't had the occasion to know the answer to the other part.

A menstrual cup would avoid problems with the flow (and it's more hygenic overall).

I go down on my lady any time. She uses tampons, so there's no mess, tho the string does get hooked in my stubble sometimes.

I've never, ever, EVER gotten any bloody discharge on me, tho I wouldn't care if I did.

She has never mentioned a weird feeling in ten years of menstrual carpet munching with tampons in, so I'll answer for her and say it doesn't feel weird.

tl;dr - no and no