Use of the subjunctive for verbs other than "to be"?

Your last example - an indirect command or request - is quite common in formal contexts, such as minutes of meetings, and even in speech for some speakers, though many would say "that he files".

The other examples are much rarer: they are archaic, and few people would use them except in special registers. But anywhere you could use "If I be", you could equally use "If he jump".

Per this NGram, although technically speaking "If I be [something], then [something]" might be (or at least, was) grammatically correct, it's not a form we use today. Best avoid it, I'd say.

I'm no grammarian, but personally I think the subjunctive form there would be "If I were wrong" anyway. I don't know what to call OP's version - all I know is if I be and if he jump sound archaic.

On the other hand, "I request that he file these papers" sounds current, if a little formal. And I'm pretty sure that is a standard example of the subjunctive. Bear in mind that, slowly but surely, use of the subjunctive mode is declining. So there are bound to be marginal cases still acceptable to some, but not to others.

The verb 'to be' is not special here.

Your examples are correct but only in a stilted, overly formal, hardly used context. In the rare context it would be incorrect to use the simple present.

But if you used those forms nowadays, it would sound strange to most English speakers of most current varieties (exceptions?).

The current pattern is to not use the subjunctive form at all and to use the simple past or present.