Hello Noel, I'm an amateur but here's how I'd approach this project.
Q1) yes you should
Q2) I use halfords grey plastic primer and have had no troubles what so ever, I use it straight from the can and it provides a very smooth uniform coat every time, but bare in mind that the colour of the undercoat/primer will affect the shade of the final coat so always check and test that the primer and top coat are going to end up the colour you want.
Q3) I would sand the emblem and lining with probably 1000 grade (or more) sandpaper, just to make sure that nothing shows through.
So there's how I'd tackle this, but perhaps an expert will offer more advice than myself,
Have fun and I hope it goes well,
Nelson
*Jinx Glenderg * (<--expert btw )