USB-C to Thunderbolt 2 (mini-Display port) to HDMI I have a monitor which I connect to an older Macbook Air using an HDMI to mini DisplayPort connector.
I also recently got a late 2016 Macbook Pro with 4 USB-C ports.

So, my question is: if I plug a USB-C to Thunderbolt 2 connector into the Macbook Pro, and then HDMI to mini DisplayPort connector into the Tunderbolt 2, would it work?

In short, would the following chain work: USB-C -> thunderbolt 2 -> mini DisplayPort -> HDMI work?
Technically speaking what you are doing is this:

    Thunderbolt 3 --> Thunderbolt 2 --> Display Port --> HDMI

USB-C and mini Display Port are interfaces while Thunderbolt, Display Port and HDMI are interfaces and signaling/protocols.

It's important to note this distinction because you cannot go from USB (A, C, or whatever) to Thunderbolt; you go from Thunderbolt to USB (it's not bi-directional).

So, assuming that you are going from TB3 to a TB2 to a mDP to HDMI, it should work because: 


TB3 is backward compatible with TB2 and TB1
TB includes the DP signaling
mDP should convert to HDMI.


That said, you may run into issues due to all the conversations.  First off, the mDP to HDMI conversion can cause issues.  It's best to just stick with one protocol like DisplayPort.  Even if you convert from mDP to DP, you are only changing the physical interface, not the protocol or signaling.

Secondly, converting from TB3 to TB2 so you can use mDP HDMI is just adding another link in the chain which becomes another point of failure/trouble.  It's always best to limit the number of connections/conversions to ensure reliability.

Finally, you didn't mention which mDP to HDMI adapter you are using, but you want to make sure you are using an active adapter (see link above for more details).  This will help with reliability of the video connection.

TL;DR

It should work, but you may run into performance/stability issues because of all the conversions.  Your best bet is to to either USB-C to HDMI or USB-C to Display Port (preferred because no conversion is required).