On the road through the woods, around the lake, out of Castelgondolfo I could hear boar but the only ones I saw close up, a couple of times, were dead on the side of a road and a railway.
Feral dogs (which are more of a problem I found in Greece last year) are not very common, but you do need to take sensible precautions when you see a flock of sheep...there will usually be some working dogs nearby and they can be quite large, but I never had a problem, even though I am not a great dog lover. They will bark a lot but if you stop, (as I have found on probably nearly 100 encounters in Italy, Albania, North Macedonia and Greece), they will not do anything more. If you then walk past slowly, I have found no problem. More helpfully, a shepherd will very often appear - always worth making friends with him. Sometimes (rarely) I have waited 20 or 30 minute for the situation to calm down!
I have never, never, never resorted to raising a stick at them nor to throwing stones. I know other people recommend this but to me it seems counterproductive.
I think this video is helpful
At Velletri, they are not strictly 'monks' - it is a novitiate for young men who are in their first year of religious life, and who are joining the Don Orione Society
https://www.facebook.com/DonOrioneVelletri/
So when I went, there had recently been a new intake and there were about 15 young men, only the minority of whom were Italian. If I recall correctly there was Mass and Evening Prayer at 1800 (to which you would be welcome, but it is not compulsory!) and then supper at 1900 or 1930. They have a very positive attitude to pilgrims with a couple of rooms set aside and one of the young men is the 'guest master' in charge of making you welcome. I think you need have no hesitation about joining them for a meal - it was like a boisterous(!) family event. You should ring ahead to say you are coming. You will need Google maps to find the place but it is on the nearside of the town, as you approach it.
I will check out where my GPS tracks came from. You can actually find my own on wikiloc.com, if you search on my name tim.redmond but they are a bit haphazardly labelled and there are 461 trails (most from running) and I have never found a good way of filtering them!