Having said that, I also have a pair of Sony MDR 7506. They are very different from the DT880s:
The Sony is closed backed, whereas the DT880 is semi open. This makes the Sony great for recording (because no sound leaks out), but when mixing the DT880 gives a much better representation of the stereo field and it makes appropriate panning and how it translates to speakers a lot easier.
Both have a more pronounced top end, but the DT880 is better at the low end. That makes mixing your bass easier on the DT880 and it also makes the Sony sound more toppy. There is still bass enough to work with, but you have to work harder at it.
In all fairness you could probably mix on both of them, once you've spent enough time with them and know their characteristics. But the DT880 make it a lot easier and are -to my ears- overall more pleasant to listen to. For any headphones, pick something that you like the sound of, that fits your budget and spend the time to get very familiar with how they sound, I'd say.
I have the 250ohm version of the DT880s and use it without a headphone amp, just plugged it in to my audio interface. It doesn’t go as loud (which is a good thing IMO, wrt ear damage) but it has been working great for me for years in this setup. Would it sound even better with a headphone amp? Probably. Can you use it without one? Definitely.
I had the 250-ohm version, which did need a headphone amp to get up to normal listening volumes. Lower impedance values means it is easier to get up to normal volumes. So, no, the 32-ohm version won't require an amp.