Issue(s) with Logitech F310 Gamepad

Hello All,
I will try to be as specific as possible. I have little to no technical expertise with computers, which is why I'm here!

I have just purchased a Logitech F310 PC gamepad so that I can play old PlayStation games, that I have just purchased very cheaply off Steam, using a controller rather than a keyboard.
Unfortunately, as long as the controller is plugged in, no in-game sound is played. Other sound is fine; I often play youtube videos and podcasts in the background while I play games and that sound is still fine even while I'm playing games. If I disconnect the F310 and go back to using the keyboard, the sound works fine.

Because the games I am trying to play are so old (the original Tomb Raider series to be specific), Xinput is not supported and I have to use Direct Input with the Logitech Gaming Software.

A secondary issue I am having is that I can't use more than one D-Pad input at a time to move Lara around. E.g. I press up and she moves forward; I press left and she turns left; if I press up + left on the D-Pad, she does nothing. I have to stop running to change directions, and then have to move in a straight line. This issue isn't so pressing as the joysticks work fine and I suppose I can get used to using them to steer!

- I have tried using a PS3 controller using SCP and Motioninjoy, and had no success with either.
- The cutscenes that play at the beginning of the game (the developer introductions and the opening sequence) play sound just fine; I assume that this is because they are pre-recorded video with audio and not sound effects of gameplay

Anyway, if you could please help me fix this issue, that would be great!

The computer is only a couple of weeks old, and this is the first time I have attempted to use the controller - the driver and software was downloaded yesterday.
It doesn't help that the Logitech Support website doesn't stop loading nothing... In either Firefox or Explorer...

No avenue of attempting to get to the Logitech Support or Feedback, or even to post a question on their forums is effective. I can't get to any of them.
Anyway, I did some more digging and tried another fix (that someone with a different controller used) which was to plug the controller in after launching the game. It keeps the sound, but I am unable to map any of the controls and it completely screws up the default controls, which cannot be changed. It's not a permanent issue, as unplugging the controller, closing the game, deleting the config.txt file, and relaunching without the controller lets me play as normal with sound with the keyboard.
Please help!

It did not come with a disk. I had to manually install the Logitech Gaming Software, which I got directly from their website.
The older Tomb Raider games have the option of either Default or User Defined controls. It will not recognise that I am using a gamepad unless I the controller is in Direct Input mode and I am running the Logitech Gaming Software. These games are too old to be compatible with Xinput.