I think they are just both in tough ages - I struggled with both of those ages so both together must be hard! I'm sure it will get easier in a few weeks when your DS starts school and DD starts being able to communicate more in the next few months.
BTW the testosterone thing is an oft-repeated myth, but a lot of 4 year olds go through a very challenging period just before they start school.
My son is 4.10 and I don't use time out because it becomes too much of a battle to get him to stay in one place. I try to manage behaviour in a related way, but if I need a single unrelated consequence then I ban him from TV for the next half hour and then I can increase it if needs be.
When cooking could you perhaps distract your son with some "big boy activity" that he can't do with his sister around, lego or colouring with some special pens she isn't allowed or something? And put her in a playpen or highchair in the kitchen so she can see you but not get out causing trouble/getting under your feet. I hated cooking with a child that age so I feel your pain. You can also try using a slow cooker which means all you have to do is chop the veg (you can do this in the morning when your older son will be at school/nursery and hopefully your DD will be napping! Or at night after they have gone to bed) and meat etc and throw it all in and switch on. Or the other thing I used to do was rely on meals like fish fingers/chicken nuggets in the oven or frozen pizza (this is good too for getting them involved as can buy a value one and let them add their own toppings) and microwave veg bags so that there were no pans on the hob and minimal hassle. Or DS likes "cold tea" especially when it is so hot, which is basically lunch - pepper, cucumber, carrot sticks, cold meats, crackers, breadsticks or bread and butter. Humous or other dips to dip the veggie sticks in are good fun too and humous is a cheap source of protein which most children like. Beans on toast or jacket potatoes is another easy and safe meal to make because you can warm the fillings up in microwave or use cold - tuna, cheese, coleslaw etc all work cold.
Lots of descriptive praise when you see DS doing something nice (so not just "good boy" but "DS that is lovely the way that you are holding DD's hand") and make sure he sees you telling DD off (mildly, even if she doesn't understand!) for things he isn't allowed to do or he might think it's unfair that she's allowed to get away with it and he isn't. This is where related consequences really work too because if someone is for example using a toy inappropriately, the toy goes away until they can behave with it. Works for both of them.
Do you think he is angry/jealous of her (possibly playing out anger at his dad being away so much?) or do you think he is trying to play but just not expressing that very well? You could try and think of some games they could play together, it's hard at her age, but something like him building a tower of bricks for her to knock down, or him rolling a ball towards her and getting her to push or kick it back. Or perhaps he could build a train track for her to push the train around or something like that. But equally it's important for him to have playtime that she isn't allowed to get into. Perhaps you can have one on one time when she naps? I think going out for days with Grandad is a great idea and it wouldn't be replacing his dad because his dad is still around.
There's a great book called "Siblings Without Rivalry" which is really good - that might have some good suggestions to help improve their relationship also?
(Just a small point too, not meaning to be rude, but your posts would be a bit easier to read if you could break them up into paragraphs every few sentences as it appears as a big block of text which can be difficult to focus on.)