When are static members a good choice in a backend class, and what problems do they cause?

A static member belongs to the class itself, not to each instance. You call it as Money.of(5, "EUR"), not on an object.

Good uses:

• Named factory methods: Money.of(), User.fromRow(row). They can validate input before creating the object, and the name says what they do.
• Constants tied to the class: Order.MAX_LINES = 50.
• Small pure helpers that need the class's private details.

Problems:

• Static state is global. A static cache = new Map() is shared by every request, every tenant and every test. One test can leak data into the next.
• Hard to replace in tests. Code that calls EmailClient.send() directly cannot receive a fake through the constructor.
• Hidden dependencies. The constructor no longer shows everything the class uses.

Inside a static method, this is the class. Subclasses inherit static members too.

In short: static is fine for factories and constants. For anything with state or side effects, use an instance and inject it.