Sadly it's easier said than done, since it's not an easy thing to fix for an existing system. We've spent quite some time improving things to ease the pain on writing tests, it was getting better but would never reach the level if we were aware of this problem in the first place - there are tens of thousand tests and we cannot rewrite them all.
I'm not too familiar with your tech stack. But there are two things you mentioned that are especially tricky to handle for testing: DB and service calls.
For DB, there are typically two ways to handle it: Use real DB, or mock it.
Real DB makes people more confident, and don't need to mock too many things. The problem is it can be slow and not parallelizable, or worse, like your case there's no impotent environment at all. We had automated migrations, but the test was run against the SQL Server on the same machine, so it was not parallelizable so the tests took more than a day to run on a single machine. On CI there are tens of machines but still takes hours to finish. In the end, we generalized things a little bit, and used SQLite for testing in a parallel manner. (Many people suggest against this because it's different from production, but the tradeoff really saved us). A more ideal approach is to have SQL sandboxing like Ecto (written in Elixir). Another ideal approach is to have in memory lib that is close to DB, for example, the ORM Entity Framework has an in-memory implementation, which is extremely handy because it's written in C# itself.
If there's no way to leverage real DB, you have to mock it. One thing that might help you is to leverage the Inversion of Control pattern to deal with DB access, there are many doctrines like DDD repositories, Hexagonal, Clean Architecture but essentially they're similar on this point. In this way, you'll have a clean layer to mock, and you can hide the patterns like EAV under those modules. As you leverage them enough, they will evolve and there would be helpers that could simplify the mocking process. According to your description, the best bet I would say is to evolve toward this direction if there's no hope on using real DBs, as you can tuck as much as domain logic into the "core" without touching any of the infrastructures. So that the infrastructure tests could be just very simple and generic.
For service calls, the obvious thing is to mock those calls. The not so obvious thing is to have well-defined service boundaries in the first place. I cannot stress this enough. When people failed to do this, they will feel they're spending a lot of time mocking services, while at the same time they feel they've tested nothing because most things are mocked. Microservices were getting too much hype over the years, but very few people pay enough attention on how to define services boundaries. The ideal microservice should be mostly independent, while occasionally calling others. DDD strategic design is a great tool for designing good service boundaries (while DDD tactic design is yet another hype, just like how people care more about Jira than real Agile, making good things toxic). We were still struggling with this because refactoring microservice is substantially harder than refactoring code within services, but we do try to avoid more mistakes by carefully designing bounded contexts across the system.
With that said, when the service boundaries are well-defined, and if you have things like SQL sandboxing, it's a breeze to test things because most of the data you're testing against is in the same service's DB, and there are very few service calls need to be mocked.