Yeah I'm pretty annoyed at that GRE score tbh, as it seems ridiculous that could pretty much relegate me straight away, whereas in the uk (where the GRE doesn't exist) I could get into a top tier school anyway...
Here it can vary, sometimes you'll be applying directly to work with a PI on a specific project for three years. Other cases, such as many welcome trust funded or other 4 year programmes, you just apply to an umbrella programme, do rotations in the first year, then pick a project to work on. Generally I think if it's a 3 year programme, I would contact the PI of the specific project you're applying to and focus the personal statement towards that work specifically. Some unis will have an online application system, others will just ask for materials (LORs, cv, personal statement ROTC) to be sent to a specific person.
cheers