1) As for the first question, as far as I know the main rule in most of situations is usus. People usually use shorten forms etc. There is nothing wrong with "Kazał mnie/mi, żeby powiedzieć Tobie/Ci", it's just longer so people don't speak like that usually. 2) "Jest frakcja, która jest zaprzysiężona by odsunąć mnie ze stanowiska" - by (or "aby", "do" etc.) is an indicator of purposefullness, intention or destiny. Btw this sentence sounds weird in Polish. Here is an example of simpler sentence in 3 versions: On jest urodzony (a)by biegać./On jest urodzony do biegania./On jest urodzonym biegaczem. 3) "Chcesz coś pić/wypić/popić?" is absolutely correct. People often ask though "czego się napijesz?" or "czy napijesz się czegoś?" 4) You can use infinitive here again, but you have to modify the sentence a bit, adding another subject: "Ona może wysłać mi swoje pracE przez e-mail, bym mógł je poprawić". If you don't modify the sentence it just doesn't make sense. 5) I don't know how to answer your question right now. Maybe someone else will be able, but I believe it's complicated. I will try to search for a compehensive answer later.

Regarding the last question, in the example you gave there is also a posibility of using the infinitive, provided that a slight modification to the sentence is made (this would have to read "skłonna trzymać samą siebie"). By the way, this sentence sounds really akward for a play... It's very simple really. If someone is "skłonny do", it must be followed by a gerundium (so you might say "mój wujek jest skłonny do picia", "jest także skłonny do zdrady"). But you can also say "jestem skłonny pomagać innym" (and the alternative here would be "jestem skłonny DO pomocy"). To sum up: Jestem skłonny do CZEGOŚ (a noun), Jestem skłonny ROBIĆ COŚ (an infinitive). I hope this helps.

Okay, the first one: The difference between kazać and poprosić: - kazać (komuś coś zrobić) - after kazać you will always have an infinitive, you can omit the person that you're telling to do X, but the infinitive will be there most of the time; - poprosić (kogoś o coś) - you rarely omit a person here but it can be done too ("proszę cię, żebyś dał mi chwilę spokoju", "proszę o chwilę spokoju"). The point is that here we can find a similar pattern that we could observe in the phrase "być skłonnym". You can only "prosić O" a noun (or a gerundium) or you can specify what you're asking for with a subclause. There is no other possibility (or I quess so). And by the way, you cannot say "Poprosił mnie, żeby powiedzieć". It is incorrect. You can only say "poprosił mnie, żebym ci powiedział". As I said, I am not a grammarian, but I can see some sort of a similarity to the English modals (I want to inf. = Chcę + inf.) and transitive verbs (ask someone to do something = poprosić KOGOŚ o COŚ). Maybe this will clear something up for you. (Because I'm confused :D)

There is really a lot of questions here. I think it's difficult for people to try to answer all of them at once. I'll try to get back to some of them later.

Here is another example. Please excuse the weird-sounding text. I am trying to improve my Polish by reading the text of the play "The Crucible" in Polish and in English. This play is about the Salem witch hunts in 17th century America. "To jest mordercza wiedźma wśród nas skłonna do trzymania siebie samej w ciemności." Here "do trzymania" is used instead of using trzymać although in English the infinitive is clearly used. I'm completely without a clue as to why.