@Ib1234567
I'm no longer active in the industry and certainly not that up to speed on training these days but FWIW :
I'll look at the Uni course link later when I have more time but to answer question 1:
Leaving aside a thing called an MPL, yes, your commercial licence when first issued usually only allows to fly (i.e. be type rated) on whatever small light twin the flying school had and that you were examined on for initial licence issue.
Go to an airline to fly bigger stuff and you will then need specific training on your final aircraft type (e.g. A320, 737) to get a type rating on that machine. That's an expensive process and some airlines actually have been known to charge for it, TBF many don't.
The Class One medical is absolutely essential before contemplating handing over any money. It's really not astronaut level or marathon runner level stuff but you can get caught out by really minor conditions which would have zero effect in other jobs but can be a complete stopper on licence issue.
Be very careful, especially if self funding, in any dealings of the Flying Training Organisations - (Flying schools, colleges)...many of them have hard nosed venture capitalists as major investors and they want your cash.
If you talk to any/read any of their info expect to hear that "it's never been a better time to train", "never been quicker to get a Long Haul Command"... they will want to sell expensive "zero to hero", residential Integrated 18 month courses, not so keen on letting you spend a bit less by spreading the course over a longer period (modular).
And yep, there's a bit of a sign that the airlines might just be starting to fund at least some funding, such as BA opening up an ab-initio scheme, as mentioned by @Oldowl .
HTH