Listen to anything you like that's well mastered and in a lossless format. Acoustic music recorded in natural spaces tends to reveal the capabilities of a transducer better than synthetic or overly close-miced recordings do, in my opinion. Classical and jazz tend to be most useful here because they tend to escape the dynamic range compression that pervades all of pop and electronic music these days, but there is well-mastered music in any genre. Any music you are super familliar with will basically tell you all you need to know really. A few albums I like to listen to when evaluating gear are Pat Metheny Group - The Way Up, Aphex Twin - Selected Ambient Works 85-92, and Miles Davis - Bitches Brew (not the remaster).

I don't think any of those other cans listed online will challenge the HD600 at all, I know the nightowl and se-master1 are both notoriously bad. But if they have an HD800, definitely go have a listen. It is a polarizing headphone and whether you like it or not will tell you a lot about your own preferences.

High headphone impedance is only an issue if the source you're listening out of has a high output impedance, if the output impedance of the source is high enough, the frequency response will be altered and there will be a FR bump around the primary resonance of the driver (usually around 100hz in most dynamic drivers) due to a lack of electrical damping. A dedicated headphone amp is always best, there are a lot of good ones out there these days that don't break the bank.

I'm not an IEM expert, I've never really felt the need for them and have never found a comfortable pair either. Over-ear designs should have more spatial realism as the sound field interacts with your whole ear (research HRTF and pinna transforms if you want to learn more) compared to IEMs which fire directly into the ear canal and so skip the HRTF entirely.

What do you think about Apple Homepods?

Haven't heard it. I know a lot of people were hyped up when it first came out but I don't really know if it delivered or not. It's supposed to be well DSP'd so I wouldnt be surprised if it beat most of the speakers in its price range though.

I'm thinking about getting them as desktop speakers, as an alternative to using headphones at work, because my ears get hot after a couple of hours, even in HD600.