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 use MP-CD1 for the projecting while charging.
Yes, You can connect the MP-CD1 with Apple iOS devices with an Apple Digital AV adapter.
Yes, Sony MP-CD1 does have the small speaker and it has audio output of 1W. We suggest using a dedicated portable speaker along with it.
Yes, Sony MP-CD1 does have the HDMI slot and you will get an HDMI cable also inside the box.
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.