As you can imagine, it doesn't take much to confuse windows 3.x So can imagine some of the more exotic hardware causing problems, but even a good basic NE2000 card wont let windows boot if it cant find what it's looking for.
The fact that sound in DOS doesn't work means this isn't a happy PC.
Fist step is to get dos working properly. If it was me I would wipe the disk and start again.
(If you want to back the original PC up its as simple file copy onto another HDD/PC whatever, the only files that wont copy across properly are the sys files but typing sys c: a: will back these up to a floppy for you)
I would install Dos then the AWE 32 drivers from the original creative installer which can still be found on their web site.
This will configure the Autoexec and config.sys files for you. the CT3980 is a PNP card so should be pretty straight forward.
Test a few games and make sure you have sound, mouse, etc.
Plug the network card back in and double check everything works as it did before.
If everything is working fine maybe back up the system here, Just dumping it onto your CF card will do for now.
Install WFW and hopefully It'll pick up the network card. 1 thing to look out for is NE2000 is a standard not an actual card, some company's cards are able to use the generic drivers better then others.
If WFW no longer boots restore the backup pre windows install and try again with no network card. If Windows starts we know its the Network card/drivers.
Installing/removing hardware in 3.x doesn't really work so always recommend a clean build after a hardware change
I'm using the AWE as I'm more familiar with it but if the Shuttle HOT-233 has a nice installer use whichever you like