There is some parking, metered areas, as well as garages. Best to Google it and plan ahead of time, or come a little early to find a spot. It is Manhattan after all.
There are some parking garages nearby. Use Google Maps to locate some options. At Amsterdam and 108th is a very affordable one ($22 for 5 hours) I used Christmas Eve (although cash only).
Parking is hard to find. There are meters but you have to circle allow 30 minutes to find a spot
There is on-street parking (free) within a block or two. Space directly in front of the cathedral is often taken by tour buses. By subway, the 1 train is a block away on Broadway.
Yes there are several parking lots on side streets. They get filled up fairly quick.
Not to my knowledge. I think public transportation would be your best bet
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