Could installing DD-WRT speed up my Internet?

As long as your router is on the compatible devices list, it certainly won't make anything worse.

That being said, even if a device does report compatibility, I have noticed on a few routers that they suffer from random wireless issues (mainly dropouts on buffalo devices).

Based on the steps you have already tried, I would first check if there are any manufactured released firmwares. If there are, try that first. If not, go for DD-WRT.

I don't think anyone can tell you for certain if the issue is a bad driver/firmware or a faulty unit as I have seen these symptoms for both and it is impossible to distinguish.