싶다/ 좋다/ 슬프다/기쁘다 etc. function as ADJECTIVES. They describe things."good/ sad' Adding 하다 change them into a VERBS. So you can use them to say 'I like'/ 'I'm sad' (1st person) You always add 하다 when talking about IT, HER, HIM, THEM... (3rd person)'

Wash rice, hands, face, body overall.

한 is a conjunction of '하다' (to do) and past particle ㄴ/은 (added to an active verb; to descriptive verb you would add 던)

Yep. Just as Park Se-june wrote. There's fixed phrase 화나다 which mean 'to be angry'. But it's often splited just like in your example. 화 itself means 'angry' and you can add 나다 (= passive verb 'to be') or 내다 (= active verb 'to get', 'to get sth'). The '냐' at the end is an interrogative ending in 해라체 (formal plain speech)

As far as I know (으)ㄴ/는다 and 라(면) are indirect quoted speech.

You look like her = [YOU]는 그녀 같이 보여 You look like Park Shin Hye = [YOU]는 박신혜 같이 보여요 Doesn't he look like David Beckham = 그는 David Beckham 같이 보이지 않아? *!!!*[YOU] and sentence final ending depend solely on who are you talking to! The pattern for "X Look like Z" is "X는 Z 같이 보이다" Use only when talking about someone outer apperance

그녀는 공부하기 바빴어요 or 그녀는 공부를 한 것이(→ 게) 바빴어요. Both sentences mean the same, it's just, as far as I know, (으)ㄴ/는 것 is more commonly used.

이뻐 (이쁘다) basically means "pretty" Frequently they use it to say e.g.: "(your daughter) has grew nicely" "nicely", "good", "pretty", "right", "well" 줘요 is often used to request (ask for) something So your example 이뻐줘요, depends on the context, could actually have different meanings. But for me, the fist thing that comes to my mind when I hear 이뻐줘요 is "pack it nicely (for me)" Hope this helps you a bit. And If I've made any mistake, someone could correct it