Copenhagen hotels

On our visit to Copenhagen we regretted that we were not staying in Nyhaven. We stayed near the train station.

Also would it be preferable to stay near the train station, the Lakes
area, or In Nyhavn?

That very much depends on your personal preference. The city centre is pretty compact and as long as you are within a reasonable walking distance you're in a great location. But staying a bit further out might also save you a bit of money, and as long you are close to a metro station, it's still easy to get around the city.

Finally, is A/C in a hotel really a necessity in Copenhagen in mid
May?

No

We stayed at the Marriott on the canal in May 2018. Great hotel and location--we walked all over the city from that location. We had air conditioning and we were thankful for it because it was in the high 90 degree range--a record setting temperature at the time, but it would normally be much cooler. From what we gathered, you wouldn't normally need air conditioning there. I remember a few people talking about staying at a hotel called Andersen Hotel and I've always remembered that. From what I heard, it sounded perfect and within walking distance of Nyhavn and much more. If we go back to Copenhagen, we will try to stay there if available.

As a local(ish) I have not had a reason to stay at any of these hotels. The areas you mention however are all quite central in Copenhagen and certainly within walking distance of each other.

Nyhavn is the most touristed of the areas you mention (and probably the most touristed area overall). For good reasons. It’s in the middle of the historic city centre, the harbour front is your neighbor and you have a large number of museums and points of interests within a very short walk. As a tourist, this is an area you are going to spend a fair bit of time, no matter where your hotel is. In that way, it checks all the boxes. But when I walk around at Nyhavn, I certainly get the impression that I’m surrounded by more international visitors than actual Danes and Copenhageners. Restaurants on Nyhavn have a reputation of being tourist traps, but you will find plenty of good establishments in the area.

AC is not needed. You’re going to a Nordic country, not a Mediterranean one. Daytime temperatues in May are typically 15-20 degrees C, and a fair bit lower at night.

We really enjoyed staying in the Nyhavn area. We liked Hotel Phoenix which is only a couple of blocks from Nyhavn. The Stromma Canal boat cruise was a wonderful way to see Copenhagen. The Nyhavn area is close enough to walk to the Rosenborg Castle, the Little Mermaid, and the Danish Resistance Museum. It is easy to take public transportation to the airport and Tivoli Gardens also. The Nyhavn Harbor area is lovely.

I have the Copenhagen Strand booked for may:

https://www.copenhagenstrand.com/

I stayed at Copenhagen Strand in July and liked it very much. It’s close to Nyhavn without being IN Nyhavn, so calmer and quieter. It’s close to a subway station so you can get most anywhere, and also walkable to many sights. The staff were friendly and helpful. The breakfast was excellent.

The room was on the small side, but I was solo so it was fine. They may have bigger rooms. They are not air conditioned, but have a tall tower fan that was quite effective. The outside temperature stayed below about 78 while I was there, but you never know.