Liens are effective according to their filing date. Since the covenants that created the HOA pre-date your original "first" mortgage, the HOA potentially could foreclose the property for non-payment of HOA fees and wipe out the your first mortgage loan. Since no lender will lend under such circumstances, the lender asks for the HOA to subordinate its prior claim on the property, if any.
Does the HOA have the right to refuse? They can demand a charge according to the by-laws, and they can demand that you bring your account current before agreeing to subordinate any liens. Both of these reasons are valid and commonplace.
Inquiring what your interest rate is not normally provided for in the CCRs. (It might be, but extremely doubtful.) Ask the person requesting the information where in the CCRs it says that you must disclose this. If it really is a rule, then talk to the board (not the management company) about changing that.
If you inspect your copy of the CCRs, you will probably (but not always) find a subordination clause about 8-10 pages into the document. This is usually incorporated to allow people to buy houses without getting a letter subordinating HOA liens every time a house is sold. If it's there, you can supply a copy of the documents to your lender so they won't need the letter at all. If it's not there, then you're back to asking for a letter.
HOAs often overstep their authority by demanding things like copies of leases for tenants. They do have a right to know if you have common facilities whom you have authorized to use those facilities, but they do not have a right to know the terms of the lease.
HOAs are run by people. They can be taught to do things properly, but most often we just grudgingly let them walk all over us. Their role is important, but they're not the neighborhood gods, despite their belief to the contrary.