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.
Sony MP-CD1 weight is just about 280 grams.
Yes, Sony MP-CD1 does have the MHL connectivity, too.
Yes, Sony MP-CD1 can work with Amazon™ fire-stick™ and you can connect external speakers for a better sound experience.
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.