I lived in Plano from 1992 to 2006. We moved to McKinney in 2006 and we love it. We chose McKinney because we wanted a neighborhood where our kids could walk to school and play outside with less supervision.
Of the three you mentioned, McKinney and Allen are similar, population-wise, but both are growing fast. Murphy is much smaller than the others.
There are three high schools in McKinney, each with a few thousand students. Allen, on the other hand, has chosen to centralize their high school students into a single school that is much, much bigger. This has both advantages and disadvantages, but schools that big might not exactly say, "small town". I know people who have moved to Murphy so that their kids could go to a smaller school.
McKinney has a charming downtown square with shops and restaurants. On weekends it is not uncommon to see local kids playing music on the sidewalks for tips. Some of the restaurants and bars also have live music fairly regularly. The downtown area also features a farmer's market set amongst historical houses and other buildings. There are often parades and festivals (hand-cranked ice cream being a summer favorite) and the annual Christmas tree lighting is a family tradition.
Compared to big cities, I'd say McKinney has a slower pace of life, however the population seems pretty active. On any given day you'll see lots of people out enjoying our city's many green spaces and hike/bike trails. On weekends it is very common to share the street with large packs of cyclists. I don't know what the actual stats are, but our city feels healthy and fit.
One thing that is consistent feedback from people moving here from other parts of the country is that they are always shocked at how friendly everyone is. I suspect this is also true for Allen and Murphy--it's not a McKinney thing, it's this part of the country. You can always count on a smile and a greeting from strangers when out walking, for example. But the character of any neighborhood is very unique. I'd recommend renting in one of the three cities and then spending some time in a neighborhood before buying if being social with your neighbors is a primary concern.
Tornadoes are a part of life in this part of the country. At least once every year the sirens will go off. Severe storms can be scary, but that's just how it goes. Tornadoes aren't specific enough to use as a differentiator on a city-by-city basis. I will say, though, that rural areas seem to get hit harder than non-rural. If you buy a trailer home out in the country you are way more likely to suffer property loss from severe weather than if you have a more structurally sound home in a more urban area. Beyond that extreme example, I'd say it shouldn't be a worry. You'll probably replace your roof at least once every ten years due to hail, but that's what homeowner's insurance is for.
You've picked a great part of the country and you've narrowed it down to three wonderful cities for families. If you want to live "out in the country" then you'll pick Murphy, otherwise, you'll go with McKinney or Allen, with the different approach to high school being a big difference between the two. Regardless which you pick, you really can't go wrong. Best of luck with your move and welcome to Texas!