Is it a requirement to keep a timer as well as splits using something like Livesplit on the side of my screen with the gameplay? Can I just keep the video footage by itself?

Check the game rules for the game you’re playing. Most don’t but you never know.

I haven't used splits. The only games I've seen splits required are a) extremely popular with history of cheating b) good time below a certain threshold. The rules will make that clear.

In games that get a handful of runs and runners a year, as in most games, you don't need to include a timer. You can eyeball the time to the whole second and submit the footage by itself. First run from a new account, it'll look suspicious but if you're new to recording and streaming then that should be understandable. Mods want more runners.

I think the easiest way to add a timer is use an online one and import the URL with OBS. Lots of tutorials. You can add one by playing the recorded video in OBS if you didn't use a timer while playing or if you need to retime.

I've seen splits required for timers that use load removers as well.