However, I can shine some light on the below:
> I would love to make myself useful for free
When I was trying to get my foot in the door, I made this offer and was always shot down. I couldn't understand why (free work is free -- is everyone insane?).
After a bit of experience (and receiving this offer regularly) I realized why my offer was always declined. The process of onboarding a dev is long and takes a toll on the existing teams productivity (it's often a worthwhile toll). The process to onboard an intern/free worker is roughly the same as a "normal" employed dev (unless the dev you're employing has identical past experience, which is very rare).
From the employer's perspective, it's often not beneficial to onboard someone that's explicitly looking for a full-time role elsewhere.