Yes, You can use MP-CD1 for the projecting while charging.
It takes about 150 minutes to fully charge the huge battery of 5000mAh of Sony MP-CD1.
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.
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.