Great work! I have been leaning in a similar direction as my girl goes to bed early IMO, and it is usually about the time I am getting motivated to jam, so a removed space would be perfect. I am thinking of getting a 20ft shipping container and working up the interior to my liking.
As for the HVAC, if you have a full system in house, you may be able to run some duct work out to that shed to utilize the house system. Otherwise, I think
@DISTORT6 summed up the options in his post. And
@Tony M. makes a very good point about not wanting to advertise what may be inside. Many will have no idea and just think you love birds, but others may use it as a reason to "case" your shed. Maybe put the birds on the interior around the door frame or window frame.
Finally, I would like to add a trick that I used on my last construction (an office extension 6ft x 7ft). Before I put up the interior walls, I took pix and mapped out everything. Wiring, outlets, studs, cross stud supports, etc. Now when I need to go into a wall or attach something thereto, I can be assured of where everything is. I know how to use a stud finder, but knowing where they are before you go searching, and how they are arranged, where cross supports and electrical are etc. has made updates a breeze!
Looking forward to progress pix!
Happy nailin'!
MW