The sound is more reference than "fun". Bass is fast and tight, but lacking in the sub bass region. Mids are pretty neutral - not warm, not cold, not forward, not recessed. Highs are pretty airy and forward with a little bit of an edge. Soundstage is small and you definitely feel like you are listening to headphones. If you like darker headphones, then this will sound bright. If you like analytical and detailed sound, you will get it, but it will be a little rough. But in the right context - on a plane watching a video or listening to music while reading a book, it's a great product.
With the AK380, it has plenty of power to drive wired. I don't recall having the volume above 80. I don't have my AK380 with me at the moment, but I generally switch to the amp module if any headphone I have requires higher than 80 volume. This was due to some increased IMD that a reviewer measured under load past 85.
As for connecting to both iPhone and AK380, that is a very important question! According to Sony's online help, it is possible. However if you read the official MDR-1000x thread, nobody has been able to make it work and the general consensus is that a firmware update may be required or it will never work. I forgot to mention that, but it's not an issue for me since I use my Apple watch for call notification. I also maybe get 1 phone call a month so not an issue for me. I will say switching from one source to another is not as simple as it can be. The quickest way that I found is to manually disconnect from source 1 by turning off BT, then power down the headphone, turn on source 2, power headphone on, manually select "connect" on source 2. What I do now is simply use my iPhone for BT connection and wired to my iPad Pro.