Hi Com!
Thought I would try to help answer some of your questions to get you started. There is a lot more involved, but hopefully this will help get you pointed in the right direction. Also, I suffer from a complete inability to write short responses, so bear with me as this will be long...
Cost:
This varies greatly; size of your TV, type of LEDs you want, what equipment you already own, etc. For me, my initial build cost me probably $300 - $350 Canadian, which is about 175 - 200 GBP. that said, I had to buy things like wire, solder & soldering iron, a multimeter, and so forth. If you own some of this stuff already, you cost will obviously be less.
Brightness:
I'll be honest in that I don't remember 100% as I've recently moved and my setup is in pieces, but I'm pretty sure you can adjust brightness in the Hyperion software - apologies if I'm mistaken.
Input Lag:
Nope! I'm a huge gamer myself and there is no input lag. the signal is either dual output from HDMI on a supporting AVR or split off from a single HDMI source but in either case, the processing occurs separately, not in between the source and display so you should not experience any gaming related input lag. I've had no issues with this myself at all.
Raspberry Pi:
The type of Pi you use can matter, but between a Pi3 and a Pi Zero, either one works. I'm using a Pi Zero W myself. The OS you choose to run on it may create some difference in how you need to configure things, but that's at your discretion, not a function of the hardware. Many people use Openelec, I personally used Raspian Lite but both are fine (and there are others).
What is a hardware consideration are the connections you'll need to make. I used a Pi Zero W as I mentioned, which came without the pin header soldered on - I found it much easier to connect my wiring to the Pi this way. I also own a Pi 2 and I Pi 3, which both came with the pin header already soldered to the board (I'm pretty sure you can get them with or without the header pre-soldered, you just have to look). This isn't a problem per se, but you may wish to give that some thought in terms of what you're most comfortable with when making that purchase.
LEDs:
Not equal. There are many to choose from, and I would suggest there's a degree of subjectivity here but the top two options in my opinion are the APA102 and the WS2812B. I think the APA102 are considered the better overall product, and I would agree with that having used both, but that's primarily form an ease-of-use perspective and again, that's for you to decide based on the following and any additional research for you personal scenario:
- APA102 cost more, sometimes twice as much depending on your options for purchase.
- APA102 use SPI protocol so they can be driven directly from the Pi (though you may need a level shifter - worry about this later)
- WS2812B are not SPI, so you'll typically need to add an Arduino to drive the LEDs (I've seen some threads, even in this forum, that there might be workarounds fo this, but I'm not familiar with them - happy to be corrected if I'm wrong)
Basically, APA102 cost more but can be much easier to work with depending on your comfort/skill level. WS2812B are often much cheaper, but usually more involved in setting them up - If you know what you're doing, this is potentially a non-issue though. I was a complete "newb" and went with the APA102 which gave me no trouble.
lastly, you'll also need to consider how many LEDs per meter you want, irrespective of which LED type you decided to use.
Grabber:
The USB grabber that you want uses the UTV007 (often called Fushicai) chipset. The only way they are "problematic" (in my experience anyway) is that sometimes when you buy from amazon third party sellers or eBay/aliexpress however, what the listing says for the chipset, and what you get aren't always a perfect match, thankfully they're pretty inexpensive.
Off/On:
Turning off the LEDs can be done in the software anytime, but there are some other solutions as well. The solutions I've seen are a bit beyond my present skill level, so I won't advise on them, but there is some great info available on this forum and generally online. Personally, I would turn my LEDs off in Hyperion and then also unplug the power anytime I wasn't using them or wasn't home for added peace of mind.
I'll stop here for now but hopefully this is helpful - Good luck!