girl+5,
You've received lots of good advice so far, and I have a few thoughts to add.....
To begin with, compliments on doing this trip on your own with five kids. I'm not sure I'd be able to do that. One thought that occurs to me is that as you'll be travelling as a single parent, you may have to pack along a notarized letter that indicates you have permission to take the kids out of the country.
As this is your first trip to Europe, I'd highly recommend reading Europe Through The Back Door prior to your trip, as that has a lot of great information on "how" to travel in Europe. Use the country guidebooks to plan sightseeing, in-city transportation, etc. Many museums, etc. are closed one day a week, and it's important to know when those days are. Also, there are "short cuts" one can take to minimize queues (ie: the Paris Museum Pass) and for summer travel you'll need to know that. It would be a good idea for many in your group to consider wearing Money Belts. Losing a Passport during your trip will not be a pleasant holiday experience!
When you format the spreadsheet for train costs, be sure to include the costs for pre-purchased trains in Italy, which can be very cheap if purchased up to 120 days in advance. However (and this is important), the fast trains in Italy have compulsory reservations and these are specific to train, date and departure time, and can only be used on the one train listed on the ticket. If you board the wrong train, you'll face hefty fines of about €50 PP which will be collected on the spot! The same fines apply to those travelling on Regionale trains (which don't have compulsory reservations), Buses or Metro with unvalidated tickets. It's important to know the rules when using public transit in Italy! I'm not as familiar with the trains in Germany and the other countries, but there do seem to be Saver Fares offered for pre-purchase. You'll have to crunch the numbers and try to figure out which method will work best in your situation. A combination of Rail pass and P-P may be best?
When you arrive in Hallstatt, you'll have to take Stefanie across the lake to the town. The fare is only ~€2 PP, so not a huge expense. You'll have changes at Attnang-Pucheim on the trips to and from Hallstatt.
It appears that you've already booked accommodations, so this is just a suggestion for future reference. Given that you'll be travelling only by public transit, I would have suggested choosing accommodations within easy walking distance to rail stations. With a group of that size, you could have also looked at staying in Hostels, as they're very cost effective. Your group would be perfect for a six-person Dorm room in a Hostel (many these days include an ensuite). That would also provide some company for your older kids to chat with. HI Hostels are well managed and clean, and good for families, but there are also some good independent Hostels. For example, one that I use in Rome has six-bed Dorm rooms (with WC) for €30 per night, per person. I don't know how that compares with the places you've found, but that's a good rate for a bed in Rome! It's close to Termini station so very convenient for getting around, lots of restaurants in the area and there's a Laundromat just around the corner.
Maybe I missed it, but one thing that's not clear from your post is how you're getting from the U.K. to Berlin? Most of your rail journeys are fairly straight forward. One other thing to keep in mind is that most cities have more than one rail station (ie: Paris has about six), so it's important to know which station you'll be using when you buy tickets, and that's also something to keep in mind when choosing accommodations.
I'd also suggest skipping Milan. I've never attended the Expo there, but you may find that there's not much there to interest your kids. It's likely going to be dreadfully crowded, and not a pleasant place to visit with five kids.