Sounds like a normal 3yo.
For 1, don't expect him to comply with verbal requests, that just gives the impression it's a choice, and no 3yo is going to want to stop playing or whatever. Take the shoes to him or take his hand/shoulders and steer him to the shoes. Never ask more than once, just state "It's time to get your shoes on" and then do it. Asking multiple times = "I don't really have to do it unless she threatens the step" or "I don't really have to do it until the 5th time". You can also try the closed choice method which is where you give two options which are both acceptable to you e.g. "Do you want to wear wellies, or shoes?"
Try motivating him to do it himself by keeping the morning routine the same every day. Consider a visual timetable so he can "see" each step - you can even laminate it and a picture of him so that he can move his picture (blu-tac) onto each step.
For 2, understand that the transition is hard, and will be worse if he's any of the following:
Tired
Hungry
Sugar crash
Hyped up
So prioritise naps (if he still has them), smooth bedtime and mealtimes and if he does not nap, your dinner-wind down-bed routine is likely to need to be absolutely solid because his energy and blood sugar levels will be at a total low in the late afternoons. If he's had a break from nursery and recently gone back this will also be very tiring for him. This might mean that any afternoon activities are a total no go for a while. If you have any which are non negotiable such as activities for the 5yo, bring snacks with protein - a cheese string, ham sandwich, yoghurt drink etc.
If you notice there is a particualr time when it's worse e.g. playdates after nursery, with one particular friend etc, put a pause on those for a while (unless you think it's worth it). To actually handle the transition, make it as simple with as few steps as possible, gather everything together yourself, let him know you'll be leaving soon (I would place hand on shoulder and make eye contact so he acknowledges this) and be prepared to carry him out under one arm if you have to!
It does pass.
Random refusal (especially when in a rush) is likely transition related as well. I sometimes find distraction e.g. talking about whatever it is we are doing next after this thing, helps shift their focus and get them interested in getting to the next activity, in which case they stop blocking whatever it is you're trying to do like get into the car. That could be guessing what's for dinner, choosing pudding, which DVD they want to watch or game they want to play, or something as silly as do you want to hop into the house like a bunny or walk like a robot?
Try to allow time. I know that's not always possible, but whenever it is, do. As you've noticed, rushing is stressful for them which causes more obstructive behaviour.
We did a lot of races at 3 as well. "Bet you can't get your shoes on before I get mine on!" "I really need a wee, I hope nobody beats me to the toilet!" Accroding to all the sibling books, don't encourage races between siblings - get them to race against an adult or something neutral like an egg timer.
Will he hold hands in certain situations e.g. when you're near a road? I found I could do OK with having DS1 walk on my left side if the road was on my right, and warn him as we got near to a crossing he had to hold hands, he'd usually comply. Or I'd say "Can I hold your hand?" Or what about one of those bikes with the parent handle? Or a backpack with strap. He might like putting his things in the backpack. Beware many coats have emergency pop off hoods, so don't use grabbing hood as an emergency stop. And if the hood is attached you can only use it if the coat is done up.