Hi I was hoping someone else would wade in and offer some advice but that hasn't happened, wee tip, people much more inclined to read and post if you space out your words.
I think you need to have a proper chat with this senior assistant. I can see why he is disgruntled if he is effectively doing your job on your non working days but not getting paid for it, so you need to find a way to differentiate what you do from what he does, and if he is doing management duties make him feel good about it i.e. "development opportunity" rather than expecting him to do your job just because he is there.
It does sound as if you are finding it hard to cover your job in 3 days a week. I won't say give up because I'd kill to go down to 3 days and would find a way to make it work . You need to be very organised, I write everything down as I go along, I'm lucky because although I work 4 days a week my manager has arranged our mgmt meeting the day I return so I don't miss out on anything.
Also it sounds like a bit of a sad fact of life that in your profession you will get the odd call on your day off and I'm probably being a bit hard here, but I think you either have to live with it, increase your hours or have everything so well documented that there is no conceivable reason to need to call you. I encourage staff to text, that way I can control when I reply to them and I'm not trying to change a nappy and juggle a phone.
Finally you sound a bit stressed by it all and a bit combatitive with the other staff, try to relax and it may not be as hard as it seems now.
Good Luck