Ok, so this is a multipart answer, but I'll try and be concise.
Firstly, there are 18 types of Safari Zones, for the 18 types, of course. Say you pick one of your friends Safari Zones at random. There are 3 starter types, so theres a 3/18, or 1/6 chance that you might be able to get a starter from the Friend Safari. But we aren't done yet!
Secondly, we have to know that the Pokemon selected are categorized in slots, and the Pokemon you get in each slot is randomized. Let's say you pick the Fire Friend Safari. So, looking at the slots, Slot 1 is useless to you, but Slot 2 has Charmeleon, and Slot 3 has Braixen. Charmeleon has a 25%, or 1 out of 4 chance of being in that specific Friend Safari, and Braixen has a 50%, or 1 out of 2 chance of being in that Safari Zone. Now lets assume that only Charmeleon is in this Safari Zone. I'm not sure if the encounter rates for Pokemon in Slot 1, 2, and 3 are equal (or as equal as they can be), so if I'm wrong on this, let me know, but for the sake of simplicity, lets say all 3 Pokemon in all 3 slots are set to 33%, so theres an approximately 1 out of 3 chance you'll find Chameleon. Now lets say theres Braixen instead of Charmeleon. The only thing that changes is that Braixen has a 50% chance of being picked for Slot 3, so instead of 1/4, you'd use 1/2.
And now, we calculate! Multiplying the 3 fractions stated above and converting it to a percent form gives you a 1.3889% chance of finding a Kanto starter or both starters, and a 2.7778% chance of finding a Kalos starter in a random Friend Safari! Of course, the more you know about the zone you go into, the more fractions you can take off, starting from the beginning, but thats my final answer. If you notice any of my math here is incorrect, let me know and I'll fix it ASAP.
And yes, you can get the Kanto and Kalos starters in custom Pokeballs. Those are the only starters/ fossil Pokemon you can get in there though.
Source: Experience, and way, way too much math than I should have done on a half-day (aaagh what was I thinking)
Hope I could help!