Why and what happens when I have negative LP?

you wont derank from -lp ever. your MMR stays the same.

you can only get -lp from dodging anyway, it just means you have extra games to play to rank up eventually.

only if you drop in MMR will you subsequently leave your rank.

if you lose a game while at a - LP, you stay at that same - LP value.

I think it just penalizes your match-making rating, more or less lowering your game's skill level.

In short, it makes it harder to climb.