There is wireless HDMI, but it can't officially be called that because it isn't licensed by the HDMI licensing bureau. It's called either Wireless HD or WHDI. DVDO makes one called the DVDO Air. That will work only in the room, it won't go through walls.
Mike is having connectivity issues with his home network and is wondering if he needs a new router. He has a desktop, three laptops, and a smart TV. Is that too many? Leo says yes and no. It depends on what he's doing. If everyone is streaming Netflix at the same time, the bandwidth won't really handle all that activity. On the other hand, if everyone is just emailing, then it should be fine. Four or five devices shouldn't be a problem. If his router is 4 or 5 years old, then it may be showing its age.
The issue is Jim's distance from the cell site. If he has line of site to it, he may be able to get an external directional antenna that would re-transmit the signal into his house. Check out CellAntenna.com. Look into the Yagi antennas. Wilson Electronics also makes cell signal boosters but he'll want to be sure he's getting an antenna that supports 4G/LTE.
Henry is wondering about something he saw that was a stick he'd plug into his computer for $99 that would give him free internet. Leo says it may exist, but there's no way it's a viable business proposition. They've got to be making their money somehow by putting ads on it.
Leo would first check to be sure that his parents haven't downgraded to cheaper internet. If it's fast enough when connected with ethernet cable, then there's something wrong with the router. Unless he's far away from the router. One of the problems with WiFi is that less than strong signal does lower the speed.
Since the instructor is currently using a handheld mic, a better solution would be a wireless lavalier mic. The challenge here is that a lavalier mic picks up more ambient noise and there is a higher potential for feedback with the PA system. It will be important to make sure there's enough distance between the mic and the speakers to avoid this. A lot of companies make lavaliers, but here's a few to choose from: