I'd rather not ask my in-laws this because they already think I'm borderline nuts for marrying their son.... And my DH slept his way through church and Sunday school so he's useless.
Question 1. If you wanted to talk to a priest about a personal issue, how would you go about doing that? I'm assuming you need to make an appointment. And where would you meet the priest - in the church, in an office connected to it?
This can vary a great deal between different parishes and priests. In a small town parish (where I now live) it's possible to just walk into the church office and meet with the priest most of the time. I have, many times. Or visit the rectory after hours or talk with him after Mass. Of course, you can make appointments and depending on his schedule, you might be encouraged to do so. In a larger urban parish (where I grew up) things might be more formal but there are also usually more than one priest in a larger parish. Most priests will talk to anyone at about any time if the situtation seems to warrant immediate interaction. As to the where you might meet with the priest, again, it's variable but any of the following would be plausible and realistic: church office, rectory, church, parish hall (often the basement), where ever the person seeking counsel feels most comfortable or where the priest will be at that time.
Question 2. How would a fairly modern priest react if a respected (apparently stable) citizen came to him and said she saw demons and wanted help? He would only vaguely know the person because she's basically a Christmas and Easter churchgoer, but they live in a small town so they've met outside of church.
It would depend on the priest. The link I'll list at the bottom of the message will offer some insight on how a priest might feel. I would hazard a guess that the majority of modern priests would look for another reason that the parishioner would see demons, such as a mental health issue, and do their best to get help for that person. Some priests, however, would consider the possibility. In some of Father Andrew Greeley's best selling novels, the issue of demons arises. One book that comes to mind is "Angels of September". A priest would also consider how reliable the person telling him about demons was and even if he did not know them well, he would consider their standing in the community.
The Catholic Church does recognize the existence of evil and there are official church rites for exorcism (seldom used, however, but they are there). Here's a prayer to St. Michael, the archangel, which delves into the defense against evil. Before the prayer, here is also what the catechism says about this prayer:
PRAYER AGAINST SATAN and THE REBELLIOUS ANGELS
PUBLISHED BY ORDER OF H. H. POPE LEO XIII
The Holy Father exhorts priests to say this prayer as often as possible, as a simple exorcism to curb the power of the devil and prevent him from doing harm. The faithful also may say it in their own name, for the same purpose, as any approved prayer. Its use is recommended whenever action of the devil is suspected, causing malice in men, violent temptations and even storms and various calamities. It could be used as a solemn exorcism (an official and public ceremony, in Latin), to expel the devil: It would then be said by a priest, in the name of the Church and only with the Bishop's permission
In the Name of the Father, and the Son, and the Holy Ghost. Amen.
PRAYER TO SAINT MICHAEL THE ARCHANGEL
Most glorious Prince of the Heavenly Armies, Saint Michael the Archangel. defend us in "our battle against principalities and powers, against the rulers of this world of darkness, against the spirits of wickedness in the high places" (Ephes. 6:12). Come to the assistance of men whom God has created to His likeness and whom He has redeemed at a great price from the tyranny of the devil. Holy Church venerates thee as her guardian and protector; to thee, the Lord has entrusted the souls of the redeemed to be led into heaven. Pray therefore the God of Peace to crush Satan beneath our feet, that he may no longer retain men captive and do injury to the Church. Offer our prayers unto the Most High, that without delay they may draw His mercy down upon us; take hold of "the dragon, the old serpent, which is the devil and Satan." Bind him and cast him into the bottomless pit "so that he may no longer seduce the nations"
Thanks!