Yes, You can use MP-CD1 for the projecting while charging.
No. Sadly Sony MP-CD1 does not have the Bluetooth™ connectivity.
Yes, Sony MP-CD1 does have a USB Type C port and you will also get a USB Type C cable inside the box.
Yes, Sony MP-CD1 comes with 5000 mAh battery and it does support the fast charging thanks to USB Type C charging slot. It takes about 2.5 hours to fully charge MP-CD1.
Yes, Sony MP-CD1 does have the MHL connectivity, too.
If you do give impromptu presentations and don't want to depend on the quality of tech you find in offices and hotels, the MP-CD1 (preferably outfitted with a small portable speaker and a wireless dongle) will do the job. If you aren't a traveling business person and you're looking for a projector with a bit more brightness, you're best served looking at full-on home theater projectors.