This is going to run long so TL;DR warning.
I have a #5 faithfull handplane (don't judge me, I didn't know any better!)
No judgement here! Two Faithfull planes formed the core of my handplane collection when I started out and I use both still regularly. I actually rate the Faithfull planes as possibly the best of the budget planes available over here, in general they have noticeably better fit and finish than Silverline planes for example (although they too can be made to work well,
see this video from Graham Haydon, Tuning Up a Cheap Handplane and note how surprised he was at how well it could be made to work).
Edit: I've just found out that Graham Haydon's videos have been removed from YouTube unfortunately.
All that waffle aside, I have some feeler guages and initially I want to check the sole of the plane for flatness. Will a piece of formica kitchen worktop be a flat enough reference surface to see how bad the plane sole is, or do I need to find something flatter (if so, what can I use that is likely to be found in an average house?)
You can put aside the feeler gauges for now. This is a 5, a jack plane, if used in its normal function it's not necessary for the sole to be absolutely flat. We should all remember that wooden planes were used for a very long time and their soles can't be expected to remain as flat as a machined piece of metal, and yet they did all that their owners required of them (and still can, there are abundant numbers of woodies of all sizes that are well over a century old and still just as capable as the day they were completed).
If you are lucky enough to buy a plane with a perfectly flat sole that's a good thing and the plane will work excellently.
However, many out-of-flat conditions on the sole of a plane are actually acceptable. Here are some examples, the coloured zones in each case indicating low spots:
I want to be very clear here, all of the above are acceptable and the plane can function well depending upon what you need it to do.
The bottom two will cause consternation with some viewers, and the conventional wisdom is that a low mouth opening is a dealbreaker, but that's not always the case. If the plane is primarily used for rougher work (as some jacks are dedicated for, also very applicable to scrub planes) the low mouth is of no consequence to the plane's function.
Note: there are numerous other issues that can cause, or contribute to, a plane not functioning well. This includes sharpness, depth of cut, the position of the cap iron/chipbreaker in relation to the edge, how well the cap iron's leading edge is fettled and the size of the mouth (if the frog is set forwards shavings can easily be blocked from rising through the throat).
Then secondly, if the sole needs a lot of flattening, is that something I can easily do on a series of grades of Scotch paper glueued to the worktop, or am I just creating more problems for myself here?
Yes you can do this using paper if you decide it's needed. But instead of regular wet-and-dry paper I would recommend that you do, at minimum, the initial bulk removal using the type of paper sold in a roll as it's very strong and you can stretch a long length of it out, this makes the lapping go very much faster if there is a lot of material to remove. So instead of stroking the plane back and forth over 25cm (10") or so you can do it over 60cm or longer (well over 2'). And if you find you have a lot to remove start at 80 grit at minimum although beginning at 60 grit would not be a problem. You can actually start and stop at 80 grit if you want to, there's no need to work up to 220 and finer as many people do but that's a discussion for another day.
Since you mention your Formica worktop, yes that should be suitable. Although many people prefer to do this kind of work on glass or a scrap piece of polished granite worktop material should be flat enough as it is made very flat initially, is inherently very strong and stable and is kept flat by the framing that supports is. Somewhere along a length of kitchen work surface there should be an area flat enough for this kind of thing, but check with the edge of a long steel rule or aluminium spirit level to confirm.
Cost is more of an issue than time...
This is one reason to prefer the roll abrasives over sheets, it's usually far cheaper for a given run of paper. It also typically lasts longer, so you save in two ways. Resin-bonded paper like this can last a lot longer than an equivalent grit of silicone carbide paper (the dark grey stuff), outlasting it by as much as a factor of 10, although good and bad examples of both types of paper are out there.
I don't want to buy a lot of equipment that could end up on ebay in 2 years.
Other than the abrasives no equipment is actually needed for flattening a plane's sole. Although you can do some of the flattening using a mill file or machinist's file (and many machinists would recommend doing so) it is quite common today for woodworkers to do the entire job on abrasive paper, cloth or film.
Beyond flattening the sole
It's important to note that the sole is possibly not the only part of a plane that might need fettling. If you want to smooth out the leading or trailing edges of the mouth, or work on the mating of the frog to the body casting, these are usually jobs for a file, so at least a flat needle file or warding file can be useful, or a mid-size machinist's file.
Files can also be used for relieving sharp edges along the sides of the sole and for rounding the leading edge at the toe and the trailing edge at the heel, although that can be done perfectly well with abrasive paper too if preferred.
Some further reading with a range of opinions and perspectives on the issue:
Plane soles should be mostly flat from Paul Sellers.
Plane and Simple by Andy King, on Get Woodworking.
Tuning a Stanley Bailey Bench Plane on Finger Lakes Guitar Repair.
Plane Soles: Ham Hands Make Iron Bananas from Chris Schwarz on Popular Woodworking.
Troubleshoot Your Plane on Popular Woodworking.