The first and the third are OK. In the first, 'What' is the pronoun that goes with without. You could re-write the sentence like this: Without what can you not leave the house. The second one isn't correct as there's no link between the sentences. You can use something like a relative clause to make it fix it: You are like my phone, without which I can't live. You are like my phone, which I can't live without. In the third one, which wall goes with against. More formally: Against which wall did you put the bed. In the fourth, it's not clear what it's impossible to root against, so you need a pronoun: When someone is nice, It is impossible to root against them. It is impossible to root against someone when they are nice.

The first one and the third one are okay. The second one needs the object pronoun "you", and the fourth one needs "him", "her" or the gender-neutral "them."

The first sentence has two equally plausible interpretations. It can mean (1) What things must you take with yourself when you leave the house? (2) What things must you keep in your house? In the second sentence, "leave" has the meaning "allow to remain". Eliminating the dangling preposition does not fix the confusion.