Datsun84300z

Hi everyone, I have an 84' Nissan Datsun 300zx 5 speed single turbo T-tops with 43k on it. It is the original motor, she's rough but coming back to life. The interior was stripped down for racing before I got it. Now I intend to keep the interior plane and keep the car as a daily fun car. I am in the process of getting it sponsored.

I just dropped $1800 in getting it set for inspection, but there's a couple issues before it can really go anywhere. First off there's a constant drain as soon as the battery is connected my taillights and door lights are on and it kills the battery. Second I'm not sure it has enough fuel pressure. Third it idles super low I'd say about under 800 rpm is it possible that the timings off a little bit? She sounds like a beast but then the kids I got it from straight piped it could it just be the exhaust loss? I do know that some clamps on the turbo are loose, any and all suggestions welcome I'm new to owning a Z before this I had an 84 Rx7 with wankel rotary so yea fun car but it's nice to have wheels with power now haha, my main goal is to get this car sound and solid then to start building it up more. I look forward to learning from this forum.