Actually, only "I'm taller than he (is)" is the only correct one. HOWEVER, most people use the other one - and they're wrong. So, you have a choice - to be grammatically correct or to use it like most people do. I prefer the correct way. It is my opinion that as long as you know the correct way, and can intentionally use it the wrong way (colloquially or otherwise), you're golden! Enjoy!!

Makeup and make-up are both correct, but mean completely different things. You apply makeup to your face. When you miss a test in school, you may have to do a make-up test. You use the hyphen here, because it is an adjectival figure of speech - it modifies "test." If it's not used as a modifier here, you drop the hyphen: " I have to make up the test." Finally, not to confuse you, if you create or invent something, you "made it up." For example: "I'm going to make up a sentence to explain the different uses of 'make up' to you." Online is correct NOW. Twenty years ago, when the Internet was new, the concept of "on-line" was fresh and new and the two words had yet to become a word. English is constantly changing, and so eventually "on-line" became an acceptable one-word idea, and the hyphen was dropped, leaving "online" as the word.

"I like summer better than winter" is perhaps a bit awkward, though not wrong. "I like summer more than winter" is definitely very typical English.