Can you lose a tier once you gain it?

In order to drop to b1 you have to have b5 MMR.

The rule is to drop whole leagues (gold to sliver or to bronze) you have to have the mmr of the lowest division. So it is possible not every likely.

Depends on what you mean by that. Do you mean lose it as in playing games and losing therefore resulting in a drop? Yes. or do you mean losing a tier as a result of not playing for the rest of the season? No. OR do you mean losing tier as a result of next season placements? Yes. However, in the case of the third scenario, you can also gain a division too if your mmr is good.

i did back in season 4 so i would say yes

Yes

Source : I used to be diamond this season.

yes

its very hard to do. ive tried to demote intentionally before and it took forever.

Yes you can, however it is quite difficult to do, as you need to lose a bunch of games while in Division 5 of a tier. You can lose drop due to inactivity/decay, however, you can only this only applies if you are Platinum or above, so you don't need to worry about that.