あげる is giving something from the speaker's point of view. くれる is someone else giving something to the speaker from the speaker's point of view. When it's in third person, e.g. someone that is not the speaker gives something to someone that is not the speaker, you look at it from the receiver's point of view. エイリアンが何かを人類にくれた Aliens gave something to humanity (This is from the receiver's point of view, someone gave them something) So in both questions, you should use くれる. Someone gave something to you? くれる. Someone gave something to someone in your family? Look at it from the receiver's eyes and use くれる. The same goes for some stranger giving something to another stranger etc.

EDIT: My bad, wrong language

"Mon" kan ikke oversættes direkte til engelsk. Det er et adverbium, der bl.a. bruges i (oftest indirekte) spørgsmål for at udtrykke usikkerhed. Det svarer lidt til "I wonder" på engelsk. Hvilken bil er mon den hurtigste? = I wonder which car is the fastest Mon det virker? = I wonder if it works Vil han mon gøre det? = I wonder if he'll do it Det bruges også i høflige spørgsmål (så de ikke virker så påtrængende) Mon du kunne række mig vandet? = Could you pass me the water? (her kan man ikke bruge "I wonder") Når "mon" bruges sammen med "ikke", udtrykker det, at noget er meget sandsynligt (mere end 50 %, efter min mening) Mon ikke han har en computer derhjemme = He probably has a computer at home Mon hun ikke ændrer sin mening = She's probably going to change her opinion "Mon ikke" (som et fast udtryk) kan bruges om noget, der er helt sikkert/helt afgjort. Smager det godt? Mon ikke! (helt sikkert!/helt afgjort!/eller udtrykket "det må du nok sige!") Jeg håber, du kan forstå, hvad jeg har skrevet, ellers kan jeg skrive det på engelsk :)

There isn't a difference in meaning, but in politeness. With こと, it's more polite. The normal あなたが好きです is more commonly used, though.

I would say "they wear hats", since the subject is plural. You could even say "they both wear hats". However, if you said "they wear a hat", most people would probably understand that they both are wearing seperate hats and not the same one :)