[E-PL3] shutter closed when lens removed + missing battery door

Hello! I recently got an Olympus E-PL3 as a donation.
It came with an empty battery and no charger, so I can't turn it on for now.

The person who gave it to me said it worked the last time they used it, but that the lens might have an issue.

Before I invest in a charger or a new battery, I have a couple of questions:

1. Is it normal that the shutter is closed when I remove the lens?
ChatGPT and another LLM claim that's normal, supposedly to protect the sensor. But when I look at photos of used E-PL3s for sale, you almost always see the sensor. So now I'm wondering if this a sign of a problem, maybe that’s what the previous owner meant by “lens issue”?

2. Missing battery door
I’ve had no luck finding a replacement, looked on eBay and Aliexpress. I found the OM System website does sell some battery covers, but not for the E-PL3. Does anyone know if the door is shared with other PEN models, or if I need to find the exact part?

Thanks in advance for any help.
I’d love to revive this little camera if it’s worth it!