Trip to Ukraine

I am planning a trip for my family (myself, my husband, and 2 kids ages 8 & 3) to Ukraine and Hungary for 2 weeks in mid-October (during my daughter’s school break). We are experienced independent travelers (20+ countries in Europe and beyond). Our children love to travel and are very adventurous when it comes to eating and experiencing new things. Last year we enjoyed 2 weeks in Hungary (Budapest, Eger, Pecs) by car. We do not require kid-oriented activities, but we do need accommodations that allow 4 people in a room and a car for longer distances.

We plan to stay in Lviv (4 nights) and 1 or 2 other locations in the Carpathians (3 nights each) then on to Hungary (Eger). However, I’m having trouble deciding between Ivano-Frankisk, Kolomyya, Chernivtsi, and Kamyanets. We plan to rent a car in Lviv and drop it off on our way to Hungary, cross the border by train or bus, then rent another car which we will return in Budapest (we will not be staying in Budapest, just catching our flight home). Right now I’m leaning towards staying in either Ivano-Frankisk or Kolomyya (day tripping from one to the other) and Chernivtsi or Kamyanets-Podisky (day trip from one to the other). I’m using the current Lonely Planet guide, but I really miss Rick’s advice on how much time to spend in a particular city (ie, is it worth a few hours, overnight, several days, etc.)

Here are my main questions:
Which of the Ukrainian towns that I’ve named would be good as a base for day trips within a 2-hour drive? Any insight into any of these towns would be much appreciated.

I know roads in general in Ukraine can be appalling, but does anyone know about roads between these particular towns?

What would be the best way to connect Lviv, this area of the Carpathians, and a train into Hungary where we could pick up a rental car?

How long does it take/how much hassle is it to cross the border from Ukraine into Hungary for US citizens with limited Ukrainian/Hungarian language skills? Is it easier by train versus bus?

Can anyone recommend a show with traditional music and dancing in either Lviv or the Carpathians? Cooking classes?

Are any other cultural experiences suitable for a family?

Thank you for your help!