I now have a 3yo DD and her behaviour is reminding me of what her big brother was like at the same age. I totally know what you are going through. I remember picking DS up under my arm during a massive tantrum at the playground, in front of all his and my friends, and walking out with tears rolling down my face because no-one else's child was being like that and he was just impossible and I didn't know what to do. I would dread non-nursery days at home with him and sometimes ended up screaming into a pillow just to get the stress out.
If you have a child who happens to be a stropper, three can be the absolute worst. They are bigger and can keep it up for longer and they also have a bigger arsenal of cunning delaying tactics and arguments. When I asked about it on here someone told me the word "threenager" which just about sums it up :)
It will not last - at least not if you stay firm and consistent now, and I know how hard that is, but it's your job to keep laying down the law over and over about what is acceptable, and carrying out consequences, as calmly and routinely as possible. Try to be matter-of-fact and say "OK, you have two minutes to calm down and stop that noise, otherwise we will have to leave" then if necessary calmly leave. When my DD hits me in anger I say "You do NOT hit" and put her in time out, and she knows she will not be getting anything she wants until she calms down, says sorry and behaves properly. But then as soon as the good behaviour is back, lots of attention, hugs and fun.
Also as they say "pick your battles" - with the getting dressed you can say "well we need to go in 2 mins, your choices are get dressed now, or you can go in your pjs/naked and we'll take the clothes with you for when you change your mind". Then do it - not in a cruel/punishing way, just "right off we go, the clothes are in my bag."
And with your dad, use the fact that DS likes him - say "I can't persuade him to get dressed, feel free to have a go, he might do it for you" and sit down and have a cup of tea.
I'm giving you the benefit of my experience but that is NOT to say I cope that well all the time - I was almost in tears again this morning with DD's latest tantrum. It's hard, really hard. Look after yourself and make time for getting a break when you can.
My DS is now 8 and lovely, really lovely, I promise!