I'm in the UK, and the specifics of an answer to this question are culture-dependent (and religion-dependent, if applicable).
It seems to me there are two separate issues here. One is the "formal" response to the bereavement by the company, and the other is your personal response as an individual.
In the UK, it would be customary for the company to organize a collection for a flower fund, or for a donation to a charity or organization which the bereaved employee might prefer as an alternative. For example I can remember one instance where the collection was for a local mountain search-and-rescue operation which the deceased was an active member of. That would be handled in a similar way to a collection for a retiree, etc. The amount of your individual donation would be anonymous, though there may be a list of the donors' signatures.
In the UK, the company would probably select a (small) number of employees to attend the funeral ceremonies as representatives of the company, not as private individuals. They would most likely be the bereaved person's manager(s) and close colleagues.
Beyond that, any other response would be entirely on the basis of personal friendship, and strangers on an internet forum can't really tell you what is appropriate for you to do.
One thing that you might like to remember (and this is based on my personal experience): the grieving process takes a long time to work through (maybe up to a year) and during that time there will almost certainly be "ups and downs" when random events trigger memories. But casual acquaintances and work colleagues will often have pretty much forgotten about the bereavement within a few days. Dealing with people who never knew, or have forgotten about the bereaved's situation, can be hard, and explaining it to strangers or reminding people about it can be a "too much information situation".
You might never get any obvious feedback from remembering to "cut the guy some slack" for as long as it takes, but do it anyway. It will mean more than a few quickly-forgotten platitudes that you might say right now.