One of the reasons I love playing the Doctor is that he is so easy for other characters to engage with xD I'm SO EXCITED.
I think he would really like Tony. Someone to talk science with but also someone who is a bit morally compromised (like him) and a little bit big for his britches (like him) but ultimately a good person who wants to help (like him!). I can see him showing up and just being all 'WHAT CAN I DO' and waving his screwdriver around.
He's not so into coffee but he'll pop by to say hello and also because coffee shops just have that feel you know. He's also always up for exploring (not really hunting, I can't imagine those gangly arms are good for a bow or anything) so if they ever need an extra hand or two or even maybe three...
Ten will be enamored with magic and appreciate Sirius' excitable nature. Also I may talk to you in the future about bringing in an Aurora Sinistra their age but...Effie first.
I can see Ten and Loki playing cards/chess/other strategy games together for some reason. Almost sort of a Master/Doctor rapport but hopefully without the trying to destroy the world aspect?