Hi there, so in little steps:
If I understood you correctly, you want to resit Maths, English language and then a selection of other GCSEs. That ought to be fine since there's only a requirement for getting a minimum grade in Maths and English GCSE in several universities. It'll be good to study the others though to give you some basic knowledge for your A-Levels (although they're probably not stricly necessary).
You can sit your GCSEs in June if you wish and think you'll be ready! Just remember to register for them with your school or an exam centre.
I honestly don't know how the universities could react to your re-sitting. Medical schools are unusually funny (in a bad way) about letting applicants resit examinations, with most med schools having some sort of restriction on resits, be that that you need to have achieved a minimum grade to be able to resit or have extenuating circumstances, which you do. However, all these policies I've only ever really read in the context of A-Level resits and so the resit policy might be a bit more lenient for GCSE and you definitely seem to have several grounds for extenuating circumstances. You should probably message the admissions department of a couple of universities for clarification.
Jobs! The care sector is perpetually hiring in this country so you shouldn't struggle to find a job. You might consider being a care assistant, in a care home or if you want something a bit more flexible, a domiciliary care assistant, since they are 0-hour contracts and so you get to set your own working hours. However, having spent this last year doing just that I think 0-hour contracts are immoral, evil stuff and that they should be outlawed. You have no guaranteed paid, you won't be paid for travel time (or not enough), and the companies that use them tend to have some questionable business practices and tend to undervalue their workers. Care work is also hard, you'll be required to do some not so nice things, and when your clients come and then go (usually to hospital, never to be seen again)... it hits you like the emotional equivalent of a brick to the face. Despite all of what I just said, it's probably your best job option. Med schools will like that you did it, the zero-hourness of it means that you can fit it around your studies (try to, say, work mornings, study afternoons; or work two, three full days a week and then study the rest of the week. Up to you really!), and despite the unpleasant things you'll have to do and the emotional burden of it, it is a truly magnificent job that'll teach you a lot about humanity, kindness and life, and it's just a nice feeling knowing that you made a difference to the people that you take care for.
Don't worry about your age. I said in some other post that age is but an illusion in young adults
Nobody will notice nor care, and uni is full of people of different ages so truly, it's not going to be a problem.
A-Levels: you need Chemistry. There are only like a grand total of 3 universities or some other crazy low number that will accept applicants without Chemistry at A-Level so you really need to take it to give you more options as to where to apply. Having dropped Psychology might look bad in many places since they like applicants that study and sit A-Levels in two years or less. But then again, you have some pretty compelling extenuating circumstances so who knows. Ask about this as well on that email you're going to send to unis asking about GCSE resits (I think you'll be fine on both accounts in enough med schools to apply due to your story but I don't know so please check with the people that do know - the unis themselves!).
Predictions: your reference provider should provide them for you. They might be OK to put whatever predicted you tell them but if they want some proof (they probably will), there are places that you can sit preliminary exams to generate predicted grades and your referee can use these to provide them.
I hope that covers everything... sorry for the long post
let me know if I missed anything. Good luck!