How likely is it to have my gaining unit sponsor me for Ranger school before I PCS there?

Your unit cannot sponsor you for Ranger School if you are not a part of that unit. If you haven't signed in, it's not your unit. That's like saying you're a Ranger because you volunteered for RASP or you're SF because you volunteered for SFAS.

However, as an infantryman, if you have at least six months till your report date, your branch will allow you to attend Ranger enroute to your next assignment. Also, if you reenlisted for Airborne and you want to walk onto Ranger School with a valid airborne physical after Airborne, your Career Counselor can easily shift your report date to your follow on station for you so you can try out for RAP week and the rest of Ranger School.

0%. You have to do the pre-ranger course before you can even get a slot at Ranger school

No one will let you go TDY enroute currently. Due to the COVID-19 policy for PCSs. Once you get to your unit they will have to start the process if they want to send you. But there are a lot of steps before that. All PCS's must be less than 10 days with no permissive TDY until you get to your gaining unit.