Thanks for providing the screenshots. From what we can see, the FX5200 successfully accelerates the browser in hardware, but does not accelerate WebGL, due to the driver version used. The situation is bleaker with the Radeon 9550, where the hardware acceleration does not work at all. I'm not surprised about the result, because those two graphics cards are too old to receive any benefit for the patch. For a Geforce FX, a 5600 is a better option since at least you have full hardware acceleration minus WebGL, which is not strictly essential.
Concerning this, there might be a way to trick Firefox into enabling full hardware acceleration, which is done by spoofing the driver version. You have to create a batch file that lets you decide the driver version and eventually, the VID and PID of your graphics card, in order to let Firefox think it's using another driver version, or even another graphics card. You can find more info about that on Google. Unfortunately YMMV with that, as the spoofer might work with some cards (for example, it does on my old 6800GT and my current X1950 Pro, as I use driver versions that are officially blacklisted by Mozilla, but they are the fastest ones for the respective cards), but it might not work with others, or even cause the browser to crash altogether.
About Youtube, as I said, you do not want to use the HTML5 player with those graphics cards, as it is already too taxing for the CPU, let alone for those poor cards, which do not have any kind of support for that feature.
There is a way for you to watch YouTube in browser much like you do with a modern computer. First, you have to download this extension, Greasemonkey:
https://legacycollector.org/firefox-addons/748/greasemonkey-3.9.xpi
Then, you must download this script called ViewTube:
http://sebaro.pro/viewtube/
Once you have downloaded it, installed and enabled it, you will have to configure VLC first, in order stream Youtube videos most likely at 360p or 480p (I used to watch videos at 480p and they ran fine on the 6800GT, I actually do not remember if I used 480p with the FX5600 though), then you have to enable the VLC plugin in Firefox, and in Viewtube, you need to set the option from the Viewtube list to "Low Definition MP4", then in the Auto dropdown menu, you must choose VLC, then click on the HD button and choose LD, and select the MD option and voila, you will be able to watch YouTube videos in browser even with a very old graphics card.
YT 2 Player does the same thing:
https://legacycollector.org/firefox-addons/550044/yt2p-1.2.1-signed.xpi
The difference between Viewtube and YT 2 Player is that unlike the former, the latter just automatically opens VLC whenever you click on a Youtube link. But first you have to configure the add-on in order to point it to the location of your VLC installation. Remember to periodically update your Youtube.lua file, or else videos will not be played.