I hope this is possible

Is it remotely possible to attack without moving the mouse up and down or side to side after clicking? Because on my computer I have limited mouse space and I find I am constantly banging my mouse on the sides of my desk.

Nope, mouse swinging is the only way.

You could get one of those roller-ball mice, where you don't move the mouse around, just roll the ball with your thumb or fingers.

Play as a mage?

I actually considered doing a magic user. But as in most Elder Scrolls games I mostly shift from being a mage to a combat oriented user very quickly

Using DOSBox, you could bind your LMB or RMB to "RMB + mouse movement in a direction", so that whenever you click it you won't need to move the mouse as well.

Ah well thank you. I'll try that now and see how that works out

You can try to adjust the mouse sensitivities in the control. I think it's in costume controls.