Just had a prospect get a 3 month injury during the playoffs. Will adding him to the DL eliminate offseason progression?
No, player will still receive off-season progression in accordance with your training budget
Will he get both injury recovery and offseason progression points?
Not exactly... You don't specifically get any Injury Recovery based on your medical, but you do get your normal progression from Training and a little bit extra from Training to make up for the injury. It's a clumsy wording but I don't know exactly how to say it. Whether your guy gets hurt on Day 1 of the season or the last day of the season and whether he's on DL or not on DL, they get a little bit back as part of rollover (training)
if not, what are opinions on DL vs leaving him hurt through offseason.
It doesn't really matter but there's no reason not to. There's definitely no downside, and there could be extra bonus that you're getting so you might as well