You are asking some great questions, but it’s very hard to give you definitive answers with so many unknowns such as where your home is located, and the overall tightness and energy efficiency of your home. i.e., good windows and doors, lot’s of shade trees, and stuff like that.
Also, not sure how the condition of your existing HVAC system. It sounds like you’re on gas heating now.
My first thought was what about a hybrid, where you add a mini-split to the new addition and call it a day.
Mitsubishi mini’s are terrific. They are legit if installed properly. The heat pumps (I personally own 5) are very efficient in the right climates. They are intended to stay on all the time and they adjust to load requirements constantly rather than starting and stopping. They are extremely quiet. They are reliable and there are more and more qualified dealers out there to service them.
Not sure how to answer the noisy air handler in the closet question. Perhaps you can go into an attic or two by visiting some friend’s place and have a listen to the running unit. I will say better air handlers are quieter overall but that is very subjective. RVs and mfg homes put air handlers in closets as do many apartment complexes. I guess you get used to the hum?
I cannot address the elec heat pump vs nat gas question. My guess is the natural gas is probably cheapest unless the mini’s are in a nice tight building envelope. The Mini’s energy consumption is very low for me where my ducted system was terribly lossy and inefficient. Like you, I didn’t have space to put good proper ductwork in my place so the mini split decision was a no brainer for me.
I have implemented many of the Mitsubishi units over 11 years, nearly all of them heat pumps. Every single customer has been thrilled with them. Most said the added cost was recouped in energy savings in <5 years and they loved the ability to give room by room climate control too.
I’m sure some others will chime in with their take and help you make a good decision. You definitely have some good options regardless.