Habitable Planet around Red Dwarf

Well, it's a relatively new field of study so I don't think there's any exact answers, but I've read up on this a bit out of interest. Red Dwarfs have 2 main disadvantages. One, which you've mentioned. They're relatively cold compared to our star so the planet would need to be comparatively close and therefor, probobly tidally locked. The other problem is red dwarfs tend to have very active sunspots compared to our sun, so the planet would need a crazy strong magnetic field to protect the planet. Neither of those are "deal breakers" but perhaps less likely than a sun-like star.

A tidally locked planet wouldn't necessarily have a scorching hot side and a freezing cold side. Modeling suggests that temperature would circulate. If it had extensive oceans on the tidally locked side, for example, that could create a cloud cover and not cause intense heat. You could also have the sun side be far enough away to be temperate and the star-side icy, or a livable section in a ring around the planet.

There's also the possibility that a tidally locked planet could have winds that we would find very difficult to adapt to.

I think you'd ideally have the sun side be the habitable side for photosynthesis.

On the flip side, if the star is too large, the star's life would probobly be too short for the planet to cool down enough to be habitable for us. The life of a star is the inverse cube of the size, so a star twice the size of our sun would only have a main sequence of 1.2-1.3 billion years. When you get a bit bigger than that, there's not enough time for a planet to cool off and a solar system to get past it's heavy bombardment period.

So I'd give a red dwarf a better chance of having a habitable planet than a sun 2.5 times or larger than our sun. That, and red-dwarfs are the most common stars in the galaxy, so I think there's a probability that some red dwarfs could have habitable planets.