Which one is correct?

Some people might say that "admit to" is idiomatic and "admit" is somehow more "proper". I'm easy with either. You can't "deny to" a charge or accusation.

Me and my wife are having a ...

Me is the objective, first person singular pronoun, you ought to have written "I and my wife . . . " or "My wife and I . . . "

Turning to the Democrats, Safire gets on Bill Clinton's case, as he puts it, for asking voters to "give Al Gore and I a chance to bring America back." No one would say [give I a break], because the indirect object of [give] must have objective case. So it should be [give Al Gore and me a chance.]

Probably no "grammatical error" has received as much scorn as "misuse" of pronoun case inside conjunctions (phrases with two parts joined by [and] or [or]). What teenager has not been corrected for saying [Me and Jennifer are going to the mall]? The standard story is that the object pronoun [me] does not belong in subject position -- no one would say [Me is going to the mall] -- so it should be [Jennifer and I]. People tend to misremember the advice as "When in doubt, say 'so-and-so and I', not 'so-and-so and me'," so they unthinkingly overapply it, resulting in hyper-corrected solecisms like [give Al Gore and I a chance] and the even more despised [between you and I].

But if the person on the street is so good at avoiding [Me is going] and [Give I a break], and even former Rhodes Scholars and Ivy League professors can't seem to avoid [Me and Jennifer are going] and [Give Al and I a chance], might it not be the mavens that misunderstand English grammar, not the speakers? The mavens' case about case rests on one assumption: if an entire conjunction phrase has a grammatical feature like subject case, every word inside that phrase has to have that grammatical feature, too. But that is just false.

[Jennifer] is singular; you say [Jennifer is], not [Jennifer are]. The pronoun [She] is singular; you say [She is], not [She are]. But the conjunction [She and Jennifer] is not singular, it's plural; you say [She and Jennifer are], not [She and Jennifer is.] So a conjunction can have a different grammatical number from the pronouns inside it. Why, then, must it have the same grammatical [case] as the pronouns inside it? The answer is that it need not.

A conjunction is just not grammatically equivalent to any of its parts. If John and Marsha met, it does not mean that John met and that Marsha met. If voters give Clinton and Gore a chance, they are not giving Gore his own chance, added on to the chance they are giving Clinton; they are giving the entire ticket a chance. So just because [Al Gore and I] is an object that requires object case, it does not mean that is an object that requires object case. By the logic of grammar, the pronoun is free to have any case it wants.

http://pinker.wjh.harvard.edu/articles/media/1994_01_24_thenewrepublic.html