Safety is something you really don't need to worry about. Assuming you're in the US, Europe is infinitely safer. The main crime you will need to watch for is pickpocketing, which is easy to avoid if you take precautions. You are at virtually no risk of violence. I (midlife woman) would rather walk around by myself in the middle of the night in London or Paris than in my hometown of 100,000.
Next, and maybe I should have started here, I really think you need to trim your itinerary back. I would say this even if you were traveling as a couple; with kids you will be miserable if you try to hit all those places. Keep in mind the following:
- On your arrival day, at least some of you will likely be tired and/or jetlagged. Don't count on this day as a "day" for sightseeing purposes, even if you will be getting in early in the day.
- You or some in your family may also feel less than 100% on your first full day as you adjust to the new time zone. Plan an easy, relaxing day this day.
- Every time you change locations, you will lose at least half a day. It takes time to pack up, get to the train station or airport, check in at the airport, actual time in motion, get from the train or airport to your new hotel, and get settled. If you plan to leave one city in the morning, don't plan on starting sightseeing any sooner than after lunch.
I would absolutely nix at least two of the Riviera, Barcelona and Berlin. These are farflung locations, most of which will require flying. Unless you are able to find early morning flights, you will be unlikely to be sightseeing by early afternoon; you will lose most of a day. And it's no fun getting to the airport early with kids to wrangle. Paris to NIce is nearly 6 hours by the fastest trains and will be unlikely to be appreciably faster by air. So you are talking about losing three days of travel for a day in the Riviera, two days in Barcelona, and two days in Berlin. Don't do it. As it is, your 4 days in Paris will be 3.5, and you plan to spend two of those at Disneyland, so you'll have 1.5 days for everything else in Paris. Unless you have more ideas for Scotland than Edinburgh castle, a whiskey distillery, and a day tour of the highlands, I would pass on Edinburgh. Again, it's a lot of travel for not much show. You don't want your trip to devolve into a tour of the train stations and airports of Europe - no fun for anyone. Don't confuse "getting the most from your trip" with "going to the most cities." Strangely enough, the opposite is true. The faster you go, the more time you spend in transit and the less you see and do, and the less time places have to sink into your memory.
I suggest limiting yourselves to an absolute maximum of four locations, and three would be better. You can always line up some day trips to add variety if you start feeling restless in a city. If you choose four, make two of them fairly close together so you won't waste a lot of time in transit. Kids usually do better with less upheaval, and they will remember more if you travel slower so they have time to acclimate to a place. The most memorable part of the trip for them is likely to be little stuff that doesn't matter much to you: feeding pigeons in the park, a street musician, that sort of thing. Give them time to find those moments.
Malls in Europe are generally on the outskirts of big cities and not that easy to get to. Unless you're going to commit a half day to shopping at a mall instead of seeing the city, that's not the way to go. Harrod's in London carries something for everyone, so that could be a good stop (don't miss the gourmet food area!). In Paris, there's Galeries Lafayette, but I've never shopped for kids there so I don't know if there are toys, if that's on your agenda.
Foodies with kids. If you mean fine dining, at least in Paris, this may be something of a problem. In Paris, children are rarely seen in fine restaurants. You might chance it if yours are immaculately behaved. Be honest with yourself about your kids: I mean, stay in their chairs seated on their bottoms at all times, talk in quiet voices, no throwing food or arguing or whining or crashing silverware into plates. And good French wait service is very slow by American standards - often several minutes between courses. Dinner in a fine restaurant can easily run 2.5 to 3 hours. That's a lot to ask of a 4 year old. Instead, I would look for places where you can eat outside so the kids have some leeway. Fortunately, French food in even modest restaurants can be very good. Just avoid places near major sights, which tend to be overpriced and poor. I don't have specific recommendations for you. Crepes are a good treat and street food to boot - let the kids try Nutella or marron (chestnut), a delicious walking dessert, or get with savory fillings for a meal.
In general I like your ideas on sights, focusing on outdoorsy things rather than museums. You may want to have a backup plan if it rains, though. You might be able to see a church or two without wearing your kids out - for example Westminster Abbey and Notre Dame.
Disneyland Paris doesn't hold a candle to Disneyland in the US. If you have already been to a Disneyland here, consider going to Parc Asterix instead, a similar park that's built around a famous French comic book series. Also, check out the Luxembourg gardens, where the kids can sail toy boats in the pond.
I know this isn't on your wishlist, but if it were me taking your trip, I would choose London, Paris, and Rome. Italy is a grand place with children - Italians just love them, they will be welcomed warmly everywhere, including restaurants where the waiters may well dote on them. It is an incredible place to be with youngsters, and there's tons to see and do there as well. There's an archaeological site that serves as a cat sanctuary, the Villa Borghese (all kinds of entertainment for kids), street artists and performers in Piazza Navona, the men in gladiator costumes by the Colosseum (charging an astronomical price to have their picture taken, but you can look for free), the Al Sogno toy store in Piazza Navona that has life-sized stuffed animals, at least the times I've been there before (I've seen a donkey and a baby giraffe among others). It's easy to visit the beach if you want a day off from sightseeing - not as famous as the Riviera, but also not a 6-hour train ride away.
Getting into town - I don't know the details for London as I haven't been there in a long time, but if you take the train to Paris, as you should, you'll be right in the middle of town. A taxi would be the way to go with luggage and kids, and not particularly expensive when you consider it's split among four people. If you fly in, it also depends on which airport you fly into - there are three in Paris, at varying distances (Beauvais being the farthest but also the base for many cheap flights - take into account what it will cost to get you from the airport into town and what baggage charges will be when comparing flights to Beauvais).
Note: Buy an airfare that goes into London and home from wherever your last stop is, to save the cost and considerable inconvenience of backtracking to your original airport. It's called a Multi-City fare - don't buy two one-ways, which is much more expensive. There's an option on every air website for Multi-City.