I’m assuming you mean weight training… and if so – yes! This may be pretty obvious but you definitely want to warm up before each workout. Some of my clients think since they did a warm up an hour ago, they’re good to go for their session. But that’s not true. We need to get the blood pumping and the body warmed before we push heavy. In fact, most people can push more weight when they’ve been warmed sufficiently. So, to avoid injury and get the most out of each of your workouts start with a warmup and end, ideally, with a stretch.
If you’re talking about doing a split-routine strength training program on alternate days then my answer is yes – you’re good to go. But if you’re looking to train twice in the same day then my answer is no – there’s no need for that.
Assuming you are healthy (although I don’t like to assume, but in this case I will make an exemption for the sake of the question), the answer is yes. I would take at least a day off between workout days to give the body time to recover. The duration, intensity and frequency they all depend on your goals. Also I would do different types of exercises each day for variety and muscle stimulation. There are many ways to approach this. I would suggest to alternate heavy lifting days with functional training days (TRX, body weight, kettlebells, ropes, sandbags, etc.) to optimize not only muscle growth but also the functionality of your muscles/body.
You are saying two weight workouts in the same day? That is fine if you can put enough energy into both. If you are doing heavy lifting then you should be resting the worked muscle groups the following day. So if you do upper and lower body the same day, the next day should be a rest of the muscle groups you worked. Aerobic exercise is fine.