Hi Jamitrek
1.) As mentioned, Begur doesn't have train station but there are nearer towns/cities that do. From Begur there is a intercity bus line so your son can move around to one of these towns or to other places in Catalonia. You want line 8, which serves Palafrugell-Begur-Girona: https://www.moventis.es/sites/moventis/files/line/files/8_palafrugell-begur_juliol-agost_2025.pdf The website is: https://www.moventis.es/ca/linies-horaris/linia-autobus-8-palafrugell-begur-girona
2.) The closest train station is indeed Flaçà -pronounced "flah-sah". Important: the stress falls on the second syllable. It's on lines R1 and/or R11. Check particulars here: https://rodalies.gencat.cat/en/horaris/index.html From there you can reach Begur by bus or by taxi. Check here: https://flaca.cat/guia-del-poble/transports-publics/
3.) Moving about in Catalonia is as safe as it's most places in Europe, this is to say, it's safe but also crime exists. I've mentioned this many times, it really depends on where you’re coming from. If you're used to a super quiet town, Barcelona might feel busy and chaotic. If you're from a rougher area, it may feel pretty chill. Most issues here are non- or low-violence (like pickpocketing), but again, other types of crime do also exist.
Enjoy