Can I connect a firewire device and an HDMI display to 2015 Macbook Air with a single Thunderbolt 2 port?
You can't daisy-chain the mixer from Firewire up to display since they aren't compatible standards. You can use both devices with a single Thunderbolt 2 port, but you'll need to get a dock. OWC sells one with HDMI and Firewire, so you can plug your monitor and mixer into the dock and then one TB2 cable into your Macbook Air. There are other TB2 docks with a second TB2 port instead of Firewire, so you can use them with your existing Apple Firewire adapter.

The first commenter is incorrect on several parts. You don't need to even consider anything with Thunderbolt 3 because you don't have TB3.