In the past when there were no computers, in office or in any secured place, people sign in by signing their signature and time in while entering into the place. When they leave, they sign out by putting the time out and sign. Website followed these words (sign-in and sign-out) in the similar way when computers came.
The notebook that people sign in and sign out basically contains logs, so it's called logbook. So it's also valid to use "login and logout" instead of "sign in and sign out". But in those days, "login" and "logout" was not used much mainly because of the fact that you are signing; signing is the most important event there.
When computers came in 1980s, the security is the most important thing, so most of the operation that user do are logged in the system. There are error logs, security logs, transaction logs and many more.
So when a user is allowed into a secured website by taking the username and password, this event is also logged in the system for security reasons. So technically speaking, it's meaningful, valid and better to call this events as "login and logout" instead of "signin and out" because you don't actually sign. But many still use "sign in and sign out" as well just following the old days. But you can see here, the use of "login" peaked after 1990. Logoff is just a synonym of logout.
What great examples!
As for the last set of sentences: I'd call them just about interchangeable, except that "a lot" would be regarded as less formal. (NOAD tags "a lot" as "informal," e.g.) If I was proofreading my own writing, I'd probably change "I don't know a lot about" to "I don't know much about." Either would be fine in conversation, though.
I don't know much about Western history. {okay}
I don't know a lot about Western history. {okay in conversation, perhaps a bit informal for writing}
As for the middle set: I'd rephrase the latter sentence to read, "I don't watch a lot of TV." The first one could also be rephrased ("I don't watch very much TV"), but those two sentences have slightly different meanings: "I don't watch TV very much" would probably be interpreted as "I don't watch TV very often"; while "I don't watch very much TV" sounds more like "I don't watch many TV programs". The difference is very subtle, but I think it's existent. If I watched a half hour of TV nightly, but always the same program, I'd be inclined to say, "I don't watch a lot of TV," but not, "I don't watch TV very much." (After all, I watch nightly!)
I don't watch TV very much. {you don't watch TV very often}
I don't watch a lot of TV. {you don't watch too many TV programs}
I don't watch TV a lot. {not wrong per se, but I'd recommed one of the others}
As for the first set, that latter sentence sounds off to me. Interestingly enough, I have no problem with its inverse: "I like Japanese food a lot." However, when speaking in the negative, "I don't like Japanese food very much" sounds much more polished than "I don't like .. a lot."
I don't like Japanese food very much.
I don't like Japanese food a lot. {use the first one, not this one}
I like Japanese food very much.
I like Japanese food a lot. {in the positive, you can use either one}
I'm marveling how, even though the three pairs of examples all have the same sentence structure, my comments are different for all three. I guess this is a trickier problem than one might first expect.
Best Answer
You are looking for a job.
I don't think you would be satisfied with just a "job opportunity". If you were offered a "job opportunity", you would want to follow through until you either got the job, or did not get the job.
Similarly, a child who wants to pet a cat does not want a "Schrödinger's cat", because Shrödinger's cat has a 50% chance of being dead when the child tries to pet it. The child wants a real, live "cat".