Once I've designed a Z80 computer, it was made to be compatible with CP/M (to benefit from the software tools available for CP/M) and ZX Spectrum (for games). It was all on a single big perfboard. Please allow me to show off, 'cause it's weekend! 
Wow - that's some wiring spaghetti!

I started out on a breadboard, then moved to DIP soldering on PCBs and am now plummeting down the SMD rabbit hole into Wonderland, it seems. Or at least BGA territory. I wanted to see if I could build something more powerful than my first computer -
one of these - and it seems I've done that in spades, with the video output being the final hurdle.
- 1. I don't understand, why a less than 50kLE FPGA would be a problem. My guesstimation is that that amount should be enough for a video adapter, including a softcore Z80, too. The original Z80 was less than 10 000 transistors, so I expect a Z80 IP to fit in less than 10 kLE, but never tested one, how big is the Z80 IP? Why would, let's say a 20kLE, not be enough?
We've filled a 10kLE FPGA and there's still more we could do with the GPU, let alone all the other bits. I want to add an
AY-8910 PSG into the FPGA as well, because that's what my first computer had and I like chiptunes.

I'll probably end up adding a better, more capable PSG (Programmable Sound Generator) too. Of course, there's always the possibility of putting a softcore Z80 into the FPGA too with the room we have, but that's not really my goal.
- 2. Why adding a whole VGA? Wouldn't be easier to just control a HDMI chip (or alike), so to get video output without trying to align to the VGA standard? Were there any Z80 computer with VGA back in the days?
We started with VGA as I knew nothing about it and it was the easiest introduction to video output. As you've missed most of the last year's worth of thread, it's important that you understand that I'm not a professional programmer and my experience with electronics started with the project to build my DIY computer two years ago (nearly three now!) I didn't really know what an FPGA was until I started this thread, my soldering skills extended to the SOIC chip and even that scared me! I'm trying to learn all this on the fly, as it were. Sometimes things stick and I get them straight away, sometimes BrianHG and everyone else on the forum have to put up with me being really thick and not getting something until it's repeated fifteen times, in fifteen different ways.

- 3. Will this Z80 try to be compatible with certain computer models, so it can run old Z80 games and software?
It runs CP/M 2.2 and CP/M 3, but it's not designed to be compatible with any particular old computer, it's just designed to be as simple as possible (until this thread started!) This GPU project is
by far the most complicated and modern part of it. Whilst I'm designing a video card for my computer, we're also making it open source so that anyone building their own computer (be that Z80-based, 680x0-based, or Intel 4004-based) can use the HDL in their project.
- 4. For the questions in the OP, that missing website has many snapshots on the WaybackMachine Internet archive:
https://web.archive.org/web/20190713224608/http://searle.hostei.com/grant/z80/SimpleZ80.html
Yep, that was answered over a year ago - but thanks.

Grant Searle opened up another website after he had issues with the internet provider for the one that went down.
- 5. My guess is the request from the OP has changed during the last ~100 pages of replies, can anybody make a short brief (2-3 phrases) of the current project, please?
Well, I've pretty-much outlined most of the brief above, but basically I had a cheap Cyclone II FPGA board and was trying to find out what all this 'FPGA' stuff was about and see if I could build the FPGA computer from Grant Searle's website, my intention being to just use the VGA HDL to create a video card for my computer.
The thread quickly turned into an education on designing and building a VGA card using a Cyclone IV FPGA, which I did and it works wonderfully. We pushed on from there after I asked a question about how would graphics acceleration work and now the GPU can draw lines, rectangles, quads, triangles, circles and arcs at rates my old computer couldn't even dream of. It also has a PS/2 port for a keyboard and my little computer is completely standalone now.
We're at the stage now where I'm designing a 'version 2' of the GPU card, using a bigger FPGA to accommodate more features, with full HDMI output, proper audio output (the 1st version has a buzzer), SD card (hopefully to replace the CF card interface I created for my computer) and replacing the PS/2 port with a USB port so I can use modern keyboards, maybe a mouse, plus DDR3 RAM for even more graphics memory.