Q: New Macbook to external monitor- How to power?

A.  Only if the adapter has some power source outside.    Look at: https://eshop.macsales.com/preorder/OWC-USB-C-Dock/

B: LG, Sony, NEC, Viewsonic are all good choices.  Viewsonic is the only non-Apple model I'm aware of with Colorsync colormatching.  Nikon has monitors that also colormatch.

C: Do not get a cheap HDMI cable.   Nyrius has some decent priced HDMI cables that I have found at Best Buy.  I wouldn't get anything cheaper than their's.

A. When I connect it to an external monitor (because I want to have a larger screen for work) will the adapter that I buy for the USB-C port to HDMI (or whatever kind of connector) also charge the laptop? I'm concerned about it only having one plug.

Yes, the $79 Apple one will, because you can connect your power supply to the adapter.

http://www.apple.com/shop/product/MJ1K2AM/A/usb-c-digital-av-multiport-adapter?f node=8b

Note: the OWC one is still not shipping, as of today.