In response to your questions, "My question is this: what, other than the Eucharist, are non-Catholics not allowed to do in a Catholic church? My understanding is that they are allowed to participate in the mass in every other way (except for responsibilities given to a priest, altar server, or reader), but, for instance, are they supposed to cross themselves with holy water upon entering the church? Are they allowed to go to confession? Can non-Catholics get married in a Catholic church?
Anyone is able to attend the Mass and make the responses. However the sacraments cannot be received by someone who is not in “full communion” with the Catholic Church…ie, you are either baptised Catholic or become Catholic thru the profession of faith after a period of study, usually through the RCIA process (RCIA is the normal process for training potential converts who are adults prior to them coming into the Church)
Someone attending Mass can even come up during Communion and receive a blessing. However if you are not Catholic, you cannot receive the sacrament of Confession/Reconcillation. I believe you could meet with a priest as a “confessor” and tell of your sins, but absolution could not be given.
Yes, non-Catholics can be married in a Catholic Church, to a Catholic. However except in unusual circumstances, there would not be a Mass during the wedding if both parties were not Catholic. I am not sure of the instances when a Mass can be said during the wedding…maybe someone else could help us with that.
Use of the sacramentals (holy water, sign of the cross, etc) can be used by non-Catholics during the particular activity taking place that they may attend.
Hope this helps.