How many losses in a row does it take to go from gold 3 0lp to gold4?

The shield lasts for 3 games after advancing a division, and 10 games after advancing a tier (3 games for Master tier).

this is after you promote

if you are straight losing from lets say g2 100lp to g3 0 lp you might demote on the game that brings you below 0lp

In my personal experience after having been demoted 2 times that I can recall, it took about 5 straight losses after already being at 0 LP to demote. Both times were about 7 loss streak.

Its really about your mmr. I am on a 14 games losing streak (^: on my other account and i was on gold 2 0 lp for 10 games. I got domoted after 9 games