You're right about 처럼 and 같다 - 처럼 can't be used by itself. It must have a verb after it. As for 다고 생각하다 and 것 같다, they're pretty similar. Both can mean VI think" and can be used interchangeably in that respect. In fact, you can use both in the same construction: 것 같다고 생각하다. However, 것 같다 can also mean "It seems like..." while 다고 생각하다 can also refer to your opinion (like 저의 생각에). Keep that in mind.

'것 같다' 1. 추측 하늘을 보니 밤에 눈이 올 것 같다. 2. 애매함 엄마: 나 새옷 샀는데, 이뻐? 딸: 이쁜 것 같아. 엄마: 안 이뻐? 딸: 좀 안 이쁜 것 같기도 해. 엄마: 이쁘다는 거야? 안 이쁘다는 거야? 딸: 글쎄... 3. 겸손 아빠: 이번에 케이팝 스타 오디션에서 우승할 수 있지? 아들: 네... 그럴 것 같습니다. 아빠: 너는 노래 잘 하잖아! 아들: 네... 그런 것 같습니다만, 저도 잘 모르겠어요. * 머리로는 그 차이를 분명히 안다고 생각하는데, 구체적으로 확실하게 설명하기 힘든 것 같습니다.

~다고 생각하다 is a combination of ~다고 (which makes the sentence a quotation), and 생각하다 (verb) = to think (of something). Appropriately, it is used to express your opinion of something, which is expressed as "I think..." in English. ~는 것 같다 has a nuance of uncertainty, so it can also be expressed as "I think..." in English, but the nuance is different as you are unsure. You can think of it as "probably". 같다 is not a verb, it is an adjective (to be the same). If you want to express "to be exactly the same" -> 똑같다. In English, we can express something that is the same as another thing as "like". This apple is like an orange // This apple is the same as an orange. A~(으)ㄴ B과/와/(이)랑 같다 = A is like B // A is the same as B. You can use 같다 to express it smells like ~냄새 같다, it tastes like ~맛 같다, it looks like (you won the game) ~같이 보이다.

[noun] + 처럼: like [noun] understand this as "as" or "as if" in English. 그의 몸은 고양이처럼 유연하다 = His body is as supple as a cat's 그는 마치 왕처럼 행동한다 = He acts as if he was a King ~ㄴ 것 같다 : I have the feeling; It seems to me ~ㄴ다고 생각한다: I think Either way, it isn't very authentic. When speaking Korean, you should supress your self. We have conjugations replacing "I think", "I believe", "I suppose", etc. Or use the passive form in order to avoid those above. The easier subject omission thereby is a nice bonus in addition then. 그거 예쁘네요: conjugation version 예뻐 보여요: passive version (kind of) 예쁜 듯 해요: adjective + universal noun "듯" = I suppose it's [adejctive]