I thought one of those states (OK or FL) dropped it, but its still a thing in PA and I believe one other state. I wouldn't worry too much about the specific rotation requirements as I think they don't care if it makes sense with what residency you are doing. They accredit people who have done path for example all the time.
The resolution 42 through the AOA is stupid easy to get. You essentially have your PD sign off on your list of rotations from intern year (they have a template you can use for this) and you complete and "osteopathic educational activity". This can be one of 3 options, (1) Grand rounds/noon didactic about OMM to your residency/department (verified by PD and with an upload of your ppt), (2) attend an osteopathic conference to get 8 hrs or more osteopathic category 1A CME, or (3) publish in an osteopathic journal related to OMM (you need to upload the CME certificate). You submit all this to AOA, pay something like $75, and you get approval within a month or so. More information is available here:
Recognition of PGY1 (Residency Year 1) - American Osteopathic Association
I completed Res 42 application last year from a conference I went to in early 2020, and I had completed my intern year in 2018. So basically as long as you check boxes, it doesn't really matter when you do what, but getting the PD verification letter is probably the most cumbersome part. I imagine with COVID you could probably get some Cat 1A osteopathic CME virtually.
I did it because I'm from PA, and may go back at some point, plus I hated the idea of not being able to be licensed anywhere in the US. Its possible this may all go away, but worst case scenario, you're out a little bit of time and relatively little money compared to all the other stuff you have to pay for as a physician.
As for other DO-specific requirements by state, it is usually specific to the unrestricted/full license, not the training license. All states, as far as I'm aware, offer training licenses (or simply don't require training licenses) without any specific additional requirements. As for unrestricted license requirements, this varies wildly by state and is specific to the DO medical board (if there is one) in that state. There are a few that have ridiculous requirements like osteopathic exams or something, but I wouldn't worry about that until later.