How to buy memory - does it depend on the CPU or motherboard?

Don't mix ram. It's a bad idea. Ram has certain timings, what you see as CAS or CL is only part of the primary set. There's over 30 timing sets in all. If these don't mesh perfectly, you'll get errors, everything from games dropping to desktop, bluescreens, inability to boot at all. Combining the bandwidths will also ply havok on the memory controller. What you'll end up having to do is lower the speeds and most likely up the dram voltage to handle all 4 dimms.

The best thing is if you want 8Gb, buy a kit of 2x4gb. This maintains dual channel.

1) Yes, you'll have 2 sets of dual channel ram if you keep it that way.
2) Yes and no. You can put mixed ram like that in the board, but to run, both sets will need to default to the slowest speed which is 1333MHz.
3) Yes and no again. Pc's for quite a while now accept either dual, single or both, so you'll have single channel 4Gb dimm in #1 and dual channel 4Gb in slots #2 & #4.
4) Most likely a world of headaches. Stick to 2x4Gb and retire the 2x2Gb. You could try it to see if it works, you'd end up with 12Gb at 1333 in that case, but you'd get better performance from 8Gb @ 1600 or even 1866MHz

I bought 16Gb (2x8Gb) Patriot 1866MHz ram on eBay for $20.

1) Ram is iffy. How it works isn't. Ram works at the slowest level unless you manually OC and that has hazards of its own. So generally, #1 4Gb 1600 + #2,4 2x2Gb 1333MHz is going to run as 8Gb of 1333MHz, if it runs at all. It might be possible to OC the 1333MHz to 1600MHz, then add the 4Gb stick, and have it stable, but there's no guarantees. And if it doesn't work, you wasted your money on ram that won't do the job alone as your largest amount is now 2x2Gb in dual channel.

2) If you break down and buy a single kit of 2x4Gb 1866MHz, you'll get much better performance from that 8Gb in dual channel than using the 4Gb/2x2Gb. First it'll be a matched set in dual channel mode, vrs 4Gb in dual channel mode with 4Gb in single channel, which is @20% loss in performance on the single channel alone. Not to mention you'll be pitting 1866MHz vrs 1333MHz.

3) Your motherboard only officially supports upto 2200MHz. Faster ram might work, but there's a strong chance it will be stifled and only work at 2200MHz. If you look at those ram speeds, they all say (OC) next to them. For some that means you stand a good chance of needing cpu OC to get the memory controller enough juice to get the ram stable at that speed. Some say it simply means it's an OC ram setting, since factory default is usually 1333MHz or 1600MHz, so obviously 1866MHz means the ram is OC'd by using XMP profiles.

Yes, 1866MHz is an XMP profile. 1600 is the usual default. This is not mentioning any possible changes in dram voltage that might be necessary.

Either way, I believe it's time to retire the 2Gb sticks. They may or may not work with the new ram, but in case they don't, you'll want any new ram to be sufficient for your needs.

Windows itself uses @2Gb of ram. Anything run through Windows is only going to have access to @1-3Gb of ram. Open 20 Chrome tabs and try doing something, and you are starting to stretch ram limits, which will cause pauses or hiccups as ram now has to sort out info with higher priorities and has little extra room to shuffle it all about.

I can sympathize with budget concerns, but your current ram isn't up to the task. Time to get ram that is.

Are you using a 64 bit operating system? If you are using a 32 bit operating system, only 4GB of ram will be usable.

And I agree with the above that it is not a good idea to mix and match ram. Might work, or maybe not.

OK that's good. I didn't want you to buy some new ram, only to find it would only be recognized as 4GB max.

What Windows are you using?

XMP profiles are 1 or more steps above default. On the Jedec tables you'll find all the available speeds and timings for the ram, this is easy to find using cpu-z. Most ram of 1600 or faster uses 1600 as default, so 1866 rated ram is really 1600 with xmp #1 profile of 1866. Rarely is 1333 default and 1600 xmp #1. 1333 is usually just that, 1333 with no xmp. But you can take 1600 and downclock to 1333. You'll see in the Jedec table the timings for the ram. It'll go from 9-9-9-27 to 7-7-7-24 or some such.

And yes. If you wish to activate xmp profiles, it will need to be done in bios, you'll see under the dram settings an option for setting xmp, if there is that option on the ram itself. Otherwise it'll automatically be set to default speeds.