Taking the points in the order you raised them:
1) I agree I will need a sound system, I just hadn't got that far. If I understand you correctly, all of the source outputs (i.e. tv digital receiver box, PS4, satellite box) will go into the back of the AV receiver and there is then one feed from the AV receiver to the projector. All of these connections, I assume, will be done by HDMI cable, is that right?
2) Assuming the connection between that AV receiver and the projector is HDMI, would it not be possible to connect them with one of these:
https://thewirecutter.com/reviews/the-best-wireless-hdmi-video-transmitter/ or is there a timing/lag problem with these?
3) I do enjoy the odd game of Project Cars, but I don't see myself as a serious gamer, so maybe I don't really need top spec kit, but I would like a good system, otherwise I'm not sure it's going to be worth all the effort.
4) If I understand you correctly, to have a full sound system, I'll need the AV receiver and speakers to go with it. Will that cover everything? I currently have a plasma tv with a sound bar and woofer box. The sound from that's pretty good, will I need to go for a 5.1 set up with the AV receiver?
5) If you could give me a rough guide as to what a reasonable sound system would cost, that would be very helpful (you're welcome to message me off board if you prefer).
1) Yeah, you got the hang of it. It's not difficult to follow the basic idea.
Most of your sources will have HDMI out as standard. You might have some legacy sources on basic video (composite; the yellow phono - though the colour is irrelevant to the function), S-video (small round connector with 4 metal pins) or Component (3x phonos: red, green blue). If you have and you want to play them through the system then let me know because it changes either the wiring or the type of AV Receiver you need.
2) I can't see a good reason to waste £200-£250 on a wireless link when (a) you're going to have to lift flooring or cut in to the ceiling to run power anyway. Just run the HDMI cable, it's simpler and will just work. End of. (b) You're opening yourself up to connection and compatibility problems that you really don't want in any install, let alone one where you're watching the pennies. (c) You're watching the pennies. If you have £200 to throw away put it somewhere it's going to make a BIG difference to your system - the speakers. Oh, don't forget, you also need a projection screen; and no, the white painted wall really won't do.
3) Sorry to burst your bubble but trust me, you're about a million miles away from anything remotely close to high-end.
A £500 Benq is about the least amount of money it's possible to spend on a projector that can call itself a home cinema model. The projector market goes up to £90,000+ for the higher-end
Sim2 and
Digital Projection brand models. Yep, that's just the projector - no screen, no bracket, no cables, nothing. High-end starts at around £10K. You're not even out of budget territory by £2K. For audio electronics you could quite easily spend anything up to £4,000 on an integrated AV Receiver, and if you go for a pre/power separates combo then you could knock on the door of £50,000 with some brands without breaking a sweat. Speakers is the same story.
None of that means you can't have a great system that will put a huge smile on yours and and your family's faces. Just don't think that you're spending a king's ransom to do it. When you're already on the bottom rung of the ladder, the next step down means abandoning the project.
4 & 5) What we are looking at potentially here is very much a starter system. £500 projector, say £200 for a frame screen, £30 for a projector bracket, and then a sound system - the spec of which depends on what you need it to do besides decode audio and amplify it. You'll need speaker cable, a subwoofer lead and some HDMI interconnects as well as the long one for the ceiling.
On the sound system then you have the AV receiver and the speakers. If this can get by with just HDMI inputs and basic video processing then the entry-level products are the £349 Yamaha YHT-1810 and the £399
Pioneer HTP075 amp and 5.1 speaker kits. However, I'd pass those ones over because they both have passive subs. If you come to upgrade on the speakers - which would be the biggest improvement - the kit speakers with each system have very little real value on their own because they need a special version of an AV receiver to drive the sub. Proper home cinema subs have their own built-in amp and so the AV receivers don't have speaker outputs for the sub; the connection is on a phono lead.
What I'd go for instead for a budget solution is a mix and match package of the JBL Cinema 510 speaker kit (£199) or the Canton 75 speaker kit (£279) with the Yamaha RX-V489 at £329. That gives a combined price of £528 ~ £608. Add in speaker cable, a sub lead and HDMI cables and the total audio budget would come in at roughly £600~£680. This would stomp over any soundbar under £1000 and provide some great surround effects.
If music was important, or the above stretched the budget too far, then I'd go second-hand, but if possible I'd up the performance at the same time by choosing an older AV amp - something that would have cost £800-£1000 originally - and partner it with bookshelf speakers and a decent sub. At this level there wouldn't be a huge saving, maybe £100-£150 depending on the speakers, but it would rock.