Can opponents ask for "alerts" in bridge?
Law 20 of The Laws of Duplicate Bridge deals with review and explanation of calls. Quoting partially:

F. Explanation of Calls

During the auction and before the final pass, any
player may request, but only at his own turn to
call, an explanation of the opponents’ prior auction.
He is entitled to know about calls actually
made, about relevant alternative calls available
that were not made, and about relevant inferences
from the choice of action where these are
matters of partnership understanding. Except on
the instruction of the Director, replies should be
given by the partner of the player who made the
call in question. The partner of a player who asks
a question may not ask a supplementary question
until his turn to call or play. Law 16 may apply
and the Regulating Authority may establish regulations
for written explanations.
After the final pass and throughout the play period,
either defender at his own turn to play may
request an explanation of the opposing auction.
At his turn to play from his hand or from dummy
declarer may request an explanation of a defender’s
call or card-play understandings. Explanations
should be given on a like basis to F1 above
and by the partner of the player whose action is
explained.

...

(a) A player whose partner has given a mistaken
explanation may not correct the error during
the auction, nor may he indicate in any manner that a mistake has been made. “Mistaken
explanation” here includes failure to alert or
announce as regulations require or an alert
(or an announcement) that regulations do not
require.
(b) The player must call the Director and inform
his opponents that, in his opinion, his partner’s
explanation was erroneous (see Law 75)
but only at his first legal opportunity, which is
   (i) for a defender, at the end of the play.
  (ii) for declarer or dummy, after the final
pass of the auction.


Alert procedures and requirements are prescribed by the Regulating Authority, such as the World Bridge Federation or the American Contract Bridge League.
The ACBL has a document explaining alert procedures, as well as a document that advises directors how to deal with failures to alert.
So, to answer your questions:

Opponents do not ask for an alert, but they may ask you for an explanation of your partner's bids when it is their turn to bid or play.

An unexpected alert, or alternatively a failure to alert when expected, by your partner presents unauthorized information to you. Not only may you not take advantage of this information, you are required to avoid actions that are suggested by the information. From Law 16, which deals with unauthorized information:



B. Extraneous Information from Partner

(a) After a player makes available to his partner
extraneous information that may suggest
a call or play, as for example by a remark,
a question, a reply to a question, an unexpected (i.e., unexpected in relation to the basis of his action)
alert or failure to alert, or by
unmistakable hesitation, unwonted speed,
special emphasis, tone, gesture, movement or
mannerism, the partner may not choose from
among logical alternatives one that could demonstrably
have been suggested over another
by the extraneous information.
(b) A logical alternative action is one that,
among the class of players in question and
using the methods of the partnership, would
be given serious consideration by a significant
proportion of such players, of whom it is
judged some might select it.
When a player considers that an opponent has
made such information available and that damage
could well result, he may announce, unless
prohibited by the Regulating Authority (which
may require that the Director be called), that he
reserves the right to summon the Director later.
The opponents should summon the Director immediately if they dispute the fact that unauthorized
information might have been conveyed.
When a player has substantial reason to believe
that an opponent who had a logical alternative
has chosen an action that could have been suggested
by such information, he should summon
the Director when play ends (It is not an infraction to call the Director earlier or later). The Director shall
assign an adjusted score (see Law 12C) if he
considers that an infraction of law has resulted in
an advantage for the offender.



If partner fails to alert a bid (or otherwise gives an erroneous explanation of one of your bids), and you are the declaring side, you must call the director immediately after the last pass ends the auction. If you are the defending side, you must call the director immediately at the end of play.