Yes, You can use MP-CD1 for the projecting while charging.
Yes, Sony MP-CD1 does have the 3.5mm headphone jack.
Yes, Sony MP-CD1 does have the HDMI slot and you will get an HDMI cable also 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, You can connect the MP-CD1 with Apple iOS devices with an Apple Digital AV adapter.
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.