While there are exceptions, you can generally use any timer application you want, as long it doesn’t cover any essential gameplay.
A ton of mobile games can be played with one hand.
Also, there are PC games that can be played with only the mouse. A lot of Popcap games fit this description (e.g., Plants vs. Zombies, Peggle).
Super Monkey Ball 3D, probably other games in the SMB series (though I haven’t run any others).
Super Mario Bros. Category Extensions has a couple of one-handed categories.
Most game requests go through the support hub, but some go through different places. I ask so I can recommend the appropriate submission form.
What game is it?
If there’s no player interaction during the cutscenes then I say go ahead and skip them. You can always add another category later if you find reason to.
Generally people don’t care. You definitely don’t need to delete your previous runs.