Connecting cat 6 cables

I think we need to make sure you know the difference between a Patch Panel and a Switch.
When wiring a building, the Patch Panel is the place that all of the Cat 6 wires come into from their remote locations. The wire typically does not have plugs on the end and so it is "punched down" onto a Patch Panel. The Patch panel has Female Cat 6 sockets on it. One for each cable.

A Switch is a device that connects all of your remote cables to your router. The switch has female cat 6 sockets but has no ability to have wires "punched down"

As a result, you very likely need both the switch and the Patch Panel. The patch panel is at the central location and provides a socket interface for each cable. You would then connect short Jumper Cat 6 Cables (male-male) from the Patch Panel to the Switch. You can get around having a Patch panel by using cables with connectors but often the problem with that is that you need larger holes in the wall to pull a cable with a connector on it vs just pulling a 1/4" diameter cable. Pulling multiple cables through a channel with connectors on them is not the standard practice.

Not sure if this answered your question but thought it might help.

The 'home run' location should be the same for your RG6 coax and category cables (for LAN and phone). If you want your router to also serve as a wireless access point, take that into consideration.

Unfinished basement mechanical room - with room for additional stuff as needed - is a great spot. If your wireless signal isn't great with the WAP in that location, use an external WAP or move just the combination router/AP to the best location. Basement mechanical room is also a good spot for your distributed audio equipment and alarm panel.

Wire the bejeebus out of the house. Hard to retrofit cables.

http://cocoontech.com/wiki/Wiring_Your_New_House_101

Also you may also want to look into a Gigabit Switch instead. You are running gigabit cabling and plan on using a 100GB switch. They are very cheap. It doesn't have to be business class Cisco switches.

Not sure what you mean by Cable Heads.
At each remote location in a new construction you would bring the cable into a wall mounted box and punch down the cable to a female Cat 6 socket. Basically at same height as your 120 V wall outlets.

At the Central Location in your basement or designated location, the cables all come into that location and get punched down to the Patch Panel. Next to the Patch Panel you mount your Switch and use short Male-Male jumpers to go between each port on the switch to each socket on the Patch panel. This should all be very neat and tidy with incoming cables bundled together and hopefully labeled as to which room or location they go to. Often labeling is just a letter and number and is put at both ends of the cable to facilitate any debug at a later time. It sounds like your electrician has done that before. As mentioned by Neurorad, it is a lot easier if the location is common for your phone, internet and audio/video distribution. Just be sure to label everything.

If the wires have plugs on them they can directly to the switch.
The most common situation is for installers to pull cable without connectors and then punch those cables down to a patch bay. But if you have cables with male connectors on them then you can surely connect them directly to the switch.