There is this small church I sometimes pass. At first I though it was Catholic simply because it was named 'St. Anthony of Padua", plus the architecture looked catholic. But then I observed the sign and it said ‘Anglican Catholic’ Now I attend my own Catholic (just Catholic) church, and Im more then happy for that. But now where does the church step in with this situation? With Anglicans acknowledging the Pope and all, what would happen if by any chance a Roman Catholic attended, an Anglican Catholic mass?
What you read on the sign is (most likely) an untruth. Some Anglicans use the word “Catholic” to describe themselves, but since they aren’t truly Catholic, what you read on the sign is simply not true.
The exception, of course, is the small number of Anglican-use Catholic parishes (as other posters have already mentioned). These parishes are in fact Catholic in every sense of the word. But that doesn’t seem to be the case here.
The best (most authoritative) source to find out about that particular parish would be the local diocese. If they’re listed on the diocese webpage, or in the diocese directory of parishes, then you know they’re Catholic. You could also simply ask a local Catholic priest.
Is this the place? St Anthony of Padua in Hackensack?
saintanthonyhackensack.org/
If it is, they are not Catholic, despite the fact that they use the word to describe themselves.
To answer your questions more directly:
You may attend any services there. Catholics are free to attend non-Catholic services so long as the reason is legitimate (social obligations, special events, seeking to learn about their beliefs/practices, etc), but never as a substitute for Catholic worship.
Attending does not in any way fulfill your Sunday Mass obligation, so if you do, you still need to attend a Mass somewhere on that day.
The “masses” there are NOT valid Masses. The bread is not consecrated because there is no validly-ordained priest celebrating Mass–therefore you cannot under any circumstances receive the bread.