Hey and welcome! I am not super familiar with Dobrich, but I can answer some of your questions.
1) Going from Dobrich it will also be easy to get to Veliko Turnovo, which is a great medium sized city for tourism. Around Varna, I highly recommend going to Pobiti Kamani, which you can take a bus to from the Varna Natural History Museum (in the sea garden). I can also recommend Tryavna and Gabrovo, but those are more difficult to get to than Veliko Turnovo. (They can both be reached by train, but require a few transfers).
2) The train website is bdz.bg Buses are a little more difficult because they are all ran by independent companies. I have yet to find a really exact bus website, and tend to just ask around when I get to a bus station (of course, that requires you have some flexibility).
3) In Sofia you might be able to use Uber, there used to be but the taxi companies are trying to regulate them away. In any smaller city you will need to figure out which local company you can trust. I have the numbers of a few taxi drivers I trust and call them directly. So when you get a good taxi driver (especially if they speak english) you can ask for their number to call them directly.
4) How much you spend depends on the quality of life you are used to, how much you go out, what things you like to do. You can search the forum to see the 2018 cost of living survey and people answer from different villages and cities.
5) Dobrich is definitely not considered a small town in Bulgaria (it is the eighth largest city). Although I haven't been there, I expect it to have most of the conveniences of other large towns in Bulgaria- there are plenty of supermarkets to shop at, which are easiest for foreigners who don't speak Bulgarian because everything is visible and the prices are clearly marked. Lidl tends to be the cheapest, but Kaufland is also a good choice. Billa tends to be more expensive. At the smaller markets, it can be difficult to get what you need before learning Bulgarian because you may have to ask for what you want. In the summer and fall there will likely be a large farmer's market for you to buy fruits, veggies, and honey. These will usually be cheaper than the supermarkets, but again the prices are usually clearly marked. In general I only barter when buying things second hand. I've never bartered in the markets here, not sure how it would go over. But if you want to make sure you are not taken advantage of, it can be a smart idea to stick at the larger chain stores at first.
6) Paper work in Bulgaria- 1) take someone with you to translate if you don't speak Bulgarian. 2) be prepared to wait. 3) be friendly instead of entitled and sour. These things make the process much easier.
7) You will need to make sure the car you are driving is insured. (Insurance goes with the car, not the driver). You will also need to sure that the car has a vinette sticker if you are driving between towns. If you are in a town and do not leave the town limits, you do not need a vinette. But all of the connecting roads require one.
8) I have not used international health insurance because healthcare in Bulgaria is inexpensive. As an example, before I was insured I had to go to an emergency clinic in Varna, and it was 20 lev for my exam. Things were easy. My recommendation in Dobrich is to find English speaking providers before you have an emergency, so you know where to go. Again, as Dobrich is not particularly small, you should not have a problem finding English speaking caregivers.
9) No experience with theft. The emergency line is 112, though.
Good luck!