Yes, Sony MP-CD1 does have the HDMI slot and you will get an HDMI cable also inside the box.
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 does have the MHL connectivity, too.
Yes, You can use MP-CD1 for the projecting while charging.
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.
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.