If your printer has the HP ePrint technology you should not have a problem but HP Cannot guarantee compatibility if it is not an HP Printer because HP does not test or support third party products.
Yes, the HP Pavilion 11.6" Chromebook has an optional adapter that allows you to connect an HDMI cable to the micro USB charging port. This would allow output to a TV or HDMI compatible monitor. The port is called a "slimport".
No Skype doesn't have an app for chromebook. I would suggest getting a tablet instead. You can remote into a desktop for most apps but if you looking to skype a phone or tablet is usually a better device to use.
You will need to contact the manufacturer of the operating system on this HP PC in order to obtain information on whether or not Skype will be comaptible with their operating system.
The chromebook is neither. It is a chromebook but you can download apps from the chromestore. The chromestore is accessible on any computer that has google chrome web browser installed.