Yes, the double is self alerting (Manual 30.2.4 pD59).

Law 40B3 is the basis for the regulation "Prior agreement by a partnership to vary its understanding during the auction or play following a question asked, a response to a question or an irregularity committed by its own side is prohibited." (Manual pB24).  I might be wrong, but I don't interpret this as meaning that responder's actions can't depend on the meaning of the opponents' calls and plays.  That would seem to be part of the basis of *any* agreement.

Essentially your agreement is of the form "If opponents double meaning X, do A, and if they double meaning Y, do B".  And that agreement is not *varied* because of the opponent's system.  Rather a specific part of it is shown to be relevant as a result of the answer to the enquiry.  My understanding is that the regulation is intended to rule out agreements of the form "My response means X, unless I ask about the double in which case it means Y". 

The responder does not risk giving UI to the opener by asking because opener would fully expect responder to find out about the meaning of the double.  If the information about the double comes from the system card then responder and opener would each have to look at the appropriate time.  But requiring both to ask about the double when the game is played face to face seems ludicrous.

In the US, Europe, etc this problem does not arise, since any double of 1NT must be for takeout, or must be alerted as for penalties. (See Manuals for UK, ACBL, etc.)

This is part of the governing bodies interpretation, just as NZ-Aust governing bodies do NOT require the alert of doubles, redoubles, etc.

Only one party needs to inquire, and responder looking at the system card would mean that the partner had better look too, so they know which alternative they are using. In practice, the writer was penalised for asking the meaning, and advising that he would respond differently depending on the answer.

Of course, that was overseas (North Island) and we should probably draw a veil over what goes on there......

We require the alert of 1C Precision, because there is a substantial difference between 1C ACOL, or SAYC, but we have the same situation here, with a penalty double typically being 15-16+, and a takeout double considerably less. The problem disappears if we are required to alert the penalty double, as is done overseas.

If the opposition doubles partner's 1NT opening, a player can turn to their LHO and say "Yes please'" - as they could with a cuebid as the bids are self-alerting. There is no UI with that.

There could be UI if you only ask in certain circumstances - ie if you always ask if you are weak. I have noticed some players have a tendency to ask about doubles if they are weak in partner's suit .

As Peter has said, the it is illegal to vary one's actions depending on whether a question is asked. It is lawful to have different defences depending on the meaning of the bid.