Yes, although taking a Paladin with a low Charisma is idiotic if you ask me
The best Paladin abilities are based on Charisma, so if you put less that 14 in Charisma, provided you can do otherwise, you're heavily limiting your Paladin character.
About the interaction with other characters, well, Sakura and Taiga had no problem with Saber once they talked with her alone, so, her answers must have been very convincing
And it is apparent that Saber is not accustomed to taking orders. I mean, she knows her part and she always tries to impose it. I mean, she isn't just waiting orders, she acts and then Shirou just tells her off most of the time and she gets frustrated (like when she was set on escorting him to school). From her strict adherence to rules, he has to be at least Lawful, and since she specifically states that she will not use underhanded tactics to win (like hurting innocents etc.) she must be Good. Well, ok, we'll either have a LG warrior with MR or high saves, a specific moral code, two-handed feat chain, and maybe a template, or a first class (high stats) unorthodox Paladin