Sally, I walk every year in May.
Some years it has been HOT and sunny crossing the Pyrenees.
Two years ago it was blizzard conditions.
Last year it was lovely.
The same for other places along the Camino.
You can go to my blog and check alongside the right to see posts of when I'm on the Camino in May/June.
But it is my favorite time - the wildflowers - fewer people - and even the colder weather makes the walking incredible!
As for your questions.
1) How cold? Depends on where you are on the Camino and (literally) every year is different, but in general, it's not that cold. I no longer bother taking any heavy clothes. I just take a short sleeve shirt, a long sleeve silk or bamboo shirt, a fleece, and a featherweight but good windbreaker. If I layer those and still am cold, I put on my
ALTUS rain poncho. I start out layered and within 20 minutes of walking I'm peeling off layers, no matter how chilly it is. I do not take gloves. I do take a featherweight wool beanie.
2) Rain Gear? Always! No matter what time of year, I've never NOT seen rain. I have an
ALTUS which you can pick up in SJPP, Zubiri, and Pamplona at the sporting goods stores.
3) 95% of people never train at all. You train on the Camino. Start out walking slow and steady and increase stage length as you get into shape. You will never go through enough snow in May to "train." If it is snowing on the Pyrenees the day you walk, it is suggested you taxi to Roncesvalles or take the Valcarlos route. They will give you that information at the Pilgrim Office in SJPP and I suggest you listen to them. There is no other place on the Camino that you will have to worry about snow that I know of. Just one day of it.
My suggestion is that you don't overthink this. It is a nice trek, not a mountain climbing expedition.
Unless you are grossly overweight or have medical issues, I doubt you'll have any problems at all.
Here are a few links when we walked in May so you can see the photos:
http://caminosantiago2.blogspot.com/2012_05_01_archive.html
http://caminosantiago2.blogspot.com/2013/05/changing-routes.html
If you follow this thread, you'll see that I'm ALL bundled up in JUNE in Santo Domingo del Calzada.
The weather was just unpredictable in 2013.
http://caminosantiago2.blogspot.com/2014/05/to-st-jean-pied-de-port.html
This year, the walk to Orisson was sunny but windy:
The very next day, my group walked into Roncesvalles in blizzard conditions.
Here are a few photos of 2015. I didn't walk this year because of an ankle injury:
http://caminosantiago2.blogspot.com/2015/05/update-we-have-photos.html