I'm a big fan of doing a regular mobility practice as my warmup and I'm working out in the early morning hours right after waking up as well. Maybe I don't do all 40 reps (for my age)of each movement like Pavel suggests in Super Joints, but definitely some neck and shoulder movements like arm swings (I use what Scott Sonnon put out in Intu Flow) as there is mild rotation in the trunk that helps me feel loose. The cossack to forward lunge and pump stretches are all a part of my regular warm up. Not too many, but enough to feel like I am "on".
The main thing you have to remember about early morning training, aside from the fact that your joints need some extra love to get lubed up, is that your discs are their most full of fluid. Severe spinal flexion within the first hour of waking is not a good idea as there is a higher chance of disc injury. Even too much spinal extension can give folks problems so I take it easy with the pump stretch on the cobra portion. A good warmup helps to distribute that fluid back into your vascular system, but your form has to be dialed in as there is less room for error. Trying to stay neutral during swings and deads is key. The pullups after deads is harder on your grip but a great way to help decompress your spine.
After a warmup, you could program your press, swings, deads, pullups, then leg raises. It will definitely tax your grip in this order, but seems like the best order to me. With your history I'd definitely do some press-ups (cobra) or hang out on your stomach for a few minutes to help your back.