Building a new PC, £400 approx, using some old parts

Your pc may be more suitable than you think.

To answer some questions:

1. Your ram is likely DDR2 which is no longer used on modern motherboards which use DDR3.
Fortunately, DDR2 is becoming more expensive and has a decent resale value on ebay.

2. What case do you have?
Is there a eeason/desire to change it out?

2. The FX chips depend on multiple cores for performance.
In single threaded apps or games, the cores perform poorly. More so without any overclocking.
They are about 30% less efficient than modern intel haswell cores on a performance per clock basis.
Actually few games can use more than 2-3 cores.
3. If a motherboard supports a cpu, there will be negligible difference in performance.
What you get with a more expensive motherboard might be the ability to use a second graphics card for gaming(not recommended by me). You might also get the ability for record level overclocking with higher voltage components. Again, not recommended by me. All brands seem to be good.
4. When you use different sizes of ram in a dual channel configuration, the odd capacity will run in single channel mode. Not a biggie since added ram is more useful than faster ram operation. That is for intel cpus, for amd, you need faster ram for better performance.
You will likely want a DDR3 8gb kit of 2 x 4gb for your upgrade.

For your upgrade.
I would normally suggest a intel G3258 which is the fastest cpu for single thread operations, but only if you will overclock it.
My suggestion would be a i3 with a high clock rate such as the i3-4370@3.8
It is a dual core, but has hyperthreads which makes it able to dispatch 4 tasks.
A i5 has 4 full cores, but at stock, the fastest would be a more expensive i5-4690 at 3.5ghz
The very fastest at stock would be a i7-4790K @4.0 but it will be twice the price of the 4370.
Any lga1150 motherboard will support any haswell cpu.
If you buy a Z97 based motherboard, it will be compatible with future 14nm broadwell cpu's
It will also allow overclocking of "K" suffix cpu's as well as the G3258.

Your 620w psu will support a discrete graphics card as good as a GTX980. You are good there.
What brand/model do you have? If it is a poor quality unit, you might want to replace it.

I suggested that your E8400 was not that bad.
You could do your upgrade in two steps.

First upgrade the graphics card and see how you do.
A GTX750ti would be a nice performance boost, and it is a more modern Maxwell card.

The second upgrade I would do is to use a SSD for the"C" drive. It is a wonderful performance improver for everything you do. 120gb is minimum, but 240gb is getting cheap with sales now. Use your hard drive for bulk storage such as videos.
Samsung and intel would be my pick.

3.