The 'tallying' image links are broken...

I don't understand your question, sorry, and the grammar isn't right. A sentence can consist of several clauses. Did you mean 'What clauses are in this sentence?". Or are you referring to some type of clause?

English grammar is descriptive, not prescriptive, and as English is rather irregular, it can certainly be confusing. Firstly - " I have to go see the doctor." isn't general English. To me, that's rather American sounding. I would always say "I have to go to see the doctor" or "I have to go and see the doctor." Secondly, are you sure it's "Here's a..."? that sounds rather odd. Are you sure it's not "There's a...". And thirdly, "I go to swim every day." isn't completely incorrect. It's certainly less odd than "Here's a... pool". If you made it "I go there to swim every day." it is perfect. It's just that the gerund form is more common in that case.

Julie answered the question pretty well in her comment, certainly with respect to precision. In terms of gentleness, to me, it could perhaps suggest that the animation is smooth, and not jerky, or perhaps that the colours are not garish. I think Julie was also quite right that it is very open to interpretation - I think the phrase is meant to give a general idea, rather than meaning anything specific.

What do you think they would use as an alternative?

The difference between them do and doing is that 'doing' has a continuous sense, and therefore suggests a longer time period. You might use that for a process where doing that thing took a longer time. In this sense, because 'going out' isn't usually a long process, they are pretty much the same meaning, and 'go' would probably be used for simplicity. Please note that 'sb' is a dictionary-only abbreviation. Native English speakers don't use it in real language.

Neither, really. It means "working", in a technical sense. "Don't use the old recorder, it isn't operative". Something is either operative or not, whereas things can have different degrees of efficiency or effectiveness. It's not a commonly used word, as it is mostly used in technical language. It can also be a noun, meaning a person that does a certain job.

In this context, it usually means something like "I performed badly", and also usually refers to a contest or fight. In simple terms, "I can not believe that I was so badly beaten". To me, "come off it" has more of a "stop being dishonest" meaning. It is suggesting that you don't believe what the person is saying.