Broadly yes, the intent is that your grade will look “the same” on different displays each with the appropriate Output Transform. There are, of course a few caveats, which mean that a trim pass done in the intended viewing environment on a particular display type is important. But just switching Output Transform will give you a good start point.
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Where a display type has a wider dynamic range and/or gamut that the one you used when initially grading, you will see aspects of the image that it was not possible to see originally. You may well want to adjust the grade if a practical light or background sky becomes distractingly bright, or a colour distractingly saturated.
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Equally, if you do your initial grade in a wide gamut and/or high dynamic range format you are able to do things which cannot be replicated in Rec. 709. The default result of switching Output Transform may lose some of the creative intent, and it may be necessary to make slightly different choices for each scene and shot to maintain the intended look.
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Although, for example, Rec. 709 and DCI-P3 versions may be “the same” when the appropriate Output Transforms are used, because the screen size, absolute luminance, display technology (emissive vs. reflective) and viewing environment are different, the same image may be perceived differently by the audience. Therefore you should trim the two versions so that they “feel the same” when viewed separately, rather than “look the same” when viewed side by side.
Regarding round-tripping back to Premiere, I assume you are talking about sending rendered graded shots back to Premiere for finishing. If done properly there should be no problem with doing this. Finishing in a different system is a commonly used approach. It is just important to check that a suitable format and bit depth is used for renders, so that there is no significant generational quality loss in the process. Resolve renders everything in 32-bit float, whereas Premiere can default to 8-bit. Check all your settings.