The o-tsuuya is the wake, at which the body is present, its face covered by a cloth.
O-soushiki is the actual funeral service prior to the cremation, usually presided over by a Buddhist priest unless the family is strongly Shinto-oriented or Christian, but I'd say that the vast majority of people have Buddhist services, even if they're not particularly religious.
After the person's death, his/her photo is put up on the family's butsudan (Buddhist altar). If you visit the family during the mourning period, you kneel before the altar and burn a stick of incense and offer a brief prayer for the soul of the deceased.
Some families keep the person's photo on the altar permanently.
The dress code for attending a wake or funeral is a black suit for men and a black dress with a string of pearls for women. Attendees bring money (I don't know what the current suggested amount is) in a special envelope (available at any stationery store).
On the 49th day after a person's death, there's a custom called "katami-waké" in which the deceased's personal possessions are divided up among friends and family members.
Since you aren't in Japan and have evidently never lived there, the family would not expect you to know their customs. I'd say that your prompt sympathy card was greatly appreciated.
You don't send New Year's cards to people in mourning, but I'm not sure what the "statute of limitations" is on this. Does it apply to people who have lost someone since the previous New Year?
In any case, you DO send a "Thinking of you during the cold weather" card late in January in place of a New Year's card.