Yep, I've heard the exact noise you describe.

Have replaced the auxilary belt tensioner on a few to stop the noise in the past.

The screech won't be affecting the starting, it's just an associated thing.

It's more battery/starter issue that's stopping it starting cleanly from cold, however, when the starter is replaced it's worth checking the old one for metallic swarf in case of impending doom from a failing DMF.