To answer your question: YES!!!
I once did some work on someone's computer a couple of years ago with a fresh, new reinstall of Windows and everything worked perfectly, even the sound. I gave the computer back to the owner but told him to get a can of compressed air to clean out the inside... Well, he did but did something to totally mess up his audio (I think he damaged the audio chip).
I tried everything to get this sound back but it was just gone (Yes, I'm a computer tech)... so my best solution to him was to order a $10 sound card from Walmart (he wanted the cheapest possible) and after it arrived I inserted the card and installed the drivers that came on the CD and it worked like a charm.
If you did that same it would work...so don't worry. You just have to remember that when you use a 3rd party sound card that you plug into your motherboard... you MUST plug your speakers into the rear of the sound card and stop using the built in ports that came with your computer. That also means the front audio ports on the case will no longer be functional... So if you choose to use headphones you have to unplug your speakers and use the GREEN port for your headphones... unless buy headphones or a headset that plugs into your USB port. In that case, that will have it's own built in sound card and you never have to unplug your speakers. In fact, I recommend just getting a headset instead..
As for your audio issues. I think your problem could be simple to fix. It sounds like you just need to install the audio driver. Go to Device Manager and search for the sound card.. it's problem something like Realtek.. and if you see a yellow icon it means you need to install the driver for it to work. Just go the the website from you computer maker.. enter the model number and download and install the audio driver and it should work again... or use the Drivers CD that came with your computer.