"But fate is a force that's simply too powerful for humans to reckon with".

Throne of Heroes have servants from across time and realities.

Whattabout those Alter-Ego Servants from Moon Cell?

Emiya was summoned because Rin was in possession of the pendant that she would end up giving to Shirou which acted as his catalyst since he kept it. I haven't played S/R, but is there anything like that going on or is it apparently random?

Think of it like this, there are multiple timelines, you're observing one of them, which from your perspective is "the present" but from a grand scale, is still in "the past".

So, you basically missed the OG timeline (which may or may not have been pruned), so now you're getting a New Game+ essentially heh

@Misfortune's Apprentice no idea. The servant in question, Assassin, is a "Rogue" servant with no master. He was simply summoned to a particular area.