Congratulations on buying your first car!
Firstly, to clear up whether it's a Mk4 or Mk5, take a look at the headlamps. The Mk5 is the facelifted version of the Mk4, with the most obviously-updated part of the car being the front-end design. Are the headlamps the same height as the grille between them, and rounded at the outer ends? Or are they taller and more triangular with orange indicator lenses? If it's the latter, you've got a Mk5.
Secondly, the guy you work with clearly knows one of the weak spots of these Fiestas; the rear wheel arches. They do have a tendency to rust quite badly, so if they're in good shape it might be worth cleaning them and greasing them as suggested.
In terms of tyres my personal experience has been that, if you're careful, you'll find that 'summer' tyres will be fine in winter, even on snow. However, I now have two sets of wheels for my Fiesta; one with summer tyres, one with proper winter tyres. The latter do certainly make a difference to cornering grip in the dry when the temperature drops, and are grippier in a straight line in snow. But if you only plan to have one set of wheels/tyres, I don't see why Continentals wouldn't be a safe bet, although I don't have any experience with them myself. If you do opt for winter tyres, I can recommend the Firestone Winterhawk 3.
In terms of engine performance, I've just got myself a 1.3 3-door, too, to temporarily replace my 1.25 5-door. Going from the more powerful car to the less powerful one, I can't say there's actually much of a difference in hill-climbing ability. In fact, living in hilly Sheffield, I'm already finding that the 1.3 is more relaxing to drive. The 1.3 makes as much torque at around 2,000rpm as the 1.25 makes at around 2,500rpm. And the extra torque of the 1.25 isn't available until around 4,000rpm, so you have to rev that engine more to get more out of it. The fact is, they're both pretty slow by 2020 standards, so unless you plan to throw lots of money at the engine, I'd just accept that you'll have to work the gearbox a fair bit. I'd instead suggest spending a little money on servicing it and putting superunleaded or a fuel additive in it, to clean up the fuel system. That might make it run a little sweeter, and certainly won't do it any harm.
I'm not sure about your options as far as changing the instrument panel appearance goes - it's not something I've messed with myself. I take it that your car has the version with just a fuel gauge, speedometer, digital odometer and warning lights. I think I've only ever seen appearance kits for the instrument panels with a tachometer.
Finally, what kind of radio/stereo are you after? In my 5-door, I swapped the factory-fitted cassette radio for a proper Ford CD radio, which only cost about £25 off eBay and at least made it easier to play a wider range of music whilst keeping a factory-fitted appearance. If you aren't bothered about that, I believe you can get adapter plates to fit any single-DIN head unit. Presumably, you then open up your options to include bluetooth, aux. input, etc.
Whatever you decide to do, I hope you enjoy the car. I've driven a bunch of different cars, and can honestly say that nothing has come close to a Mk4/5 Fiesta. That's why I've got 2 of them! They're very easy to drive, surprisingly refined, and they handle well. Tip: use that last bit to your advantage against the lack of power - the way to drive these, like any low-powered car, is to conserve momentum. Just take it easy until you've got those new tyres and have learnt their limits!