"Check out (object)" (v.) - a somewhat informal phrasal verb / expression similar in meaning to the phrasal verb "take a look," or "echar un vistazo" in Spanish. Does not have to literally mean to "look at" but rather refers to the idea of examining something. Similar in meaning to "comprobar" in Spanish.
"Check (object)" - similar to "check out (object)" in that it also means "to look at / examine," but often with a more specific intent to verify / find out something. I.e. "to check something" = lo que se hace para averiguar algo. E.g. "Check the screens to see which gate our flight leaves out of." Whereas "Check out the screens" is more open ended; "I'm interested in what the screens look like in this airport. Let's check them out."
The other way "check out (object)" can be used is to borrow something, usually in the case of borrowing a book from a library. "I would like to check out this book, please."
"Checkout" or "Check out" (n.) - leaving someplace at which the person was staying, usually a hotel. "Checkout is at 11am." Can also be used as a verb; "What time do we check out?" No object is used. You can say, "What time do we check out OF the hotel?" But "What time do we check out the hotel?" would mean "What time are we looking at the hotel?"
"View" - literally a synonym for "look at" or "see," but usually used in the technical context / social media context, as Maya Cotton also explained. Both a verb and a noun: "5 people have viewed your account." "This video is gaining popularity so fast, it already has 1,000 views in the first hour!" (in other words, the video has been viewed 1,000 times in the first hour).
"To look at" - basically the same as "to view," but does not have as strong a connection to social media as "view." A very general meaning that can be used in many different situations. Literally "mirar".
Hope this helps!