I have inadvertently lightly carbonated more than a few Meads after bottling by not being careful when racking from fruit when doing Melomels. (No I don't inhibit the yeast prior to bottling i react to the chemical)
I agree with Maylar carbonation or not is a personal preference.
I don't care too much for the carbonation myself, however my wife and daughter love it when it does happen. So we both win when it does happen!
I'm trying to answer this very same question for myself. I just split a 3 gal pear melomel in primary after fermentation stopped at just below the yeast's tolerance. That one's gonna be my carbonated one. For the other gallon and a half, I step fed honey until the yeast quit. That one will be still. They started out in primary together with pear juice and spices, they'll be split in secondaries but both racked to fresh/frozen pears. Of course, one will be significantly higher ABV because of the step feed, but they should be similar in taste.
My rule of thumb is that stronger, high gravity meads (sweet mead, show mead, big melomels) are not carbonated or refrigerated. Lighter body meads or low alcohol meads and melomels may be carbonated and sometimes refrigerated if appropriate.
Bottle carbing meads and cysers can be done, but it's tricky.
I've carbed apple cider in 22oz. bottles by monitoring my fermentations and bottling just before the process is finished. It's not an experiment I'd recommend for safety reasons because wines and ciders can attenuate more than ales - it could lead to bursting if you're not careful.
If you're interested in carbing your products, it would be much safer and faster to learn how to keg. If you choose this method, sparkling wines and meads are easy to do because it's the same principle as draft beer.