It's no for all questions.
You're not required to put any money down.
From my knowledge few traders did trade remote but they weren't new...it's something they worked with their satelite office over time. I don't recommend that.
The deal is preatty simple. The company gives you $400K - $600K and you just need to make money, which is the hardest thing to do probably. They will limit your daily losses if you're not being consistent and give you the 'high-five' when you do.
I don't know if you're new or not, prop trading is a very tough job to make a living. It's one of the best jobs ever, but 8 out of 10 traders leave within 1-2 years without making more than 30-40K a year.
Keep that in mind and good luck!