Awake has two meanings:
As an adjective it describes a person or animal's state. It may only be used as a predicate adjective, in the predicate of a clause, not as an attributive adjective before a noun:
ok John is awake and at work.
∗ Awake John is at work.
As a verb it is intransitive—it takes no object—and means “to become awake (adj)”. It is an irregular verb, with the past form awoke; two different past/past participles are in use, awoken and awaked:
I awake at four o'clock most mornings.
I awoke at six o’clock this morning.
I have awoken/awakened early only twice this month.
Awaken is a transitive verb requiring a direct object. It means “cause [someone or something] to become awake”. It is a regular verb: both the past and past participle forms are awakened:
My wife awakens me if I oversleep.
My wife awakened me at seven o’clock.
He was awakened by a loud crash.
The verb wake, woke, woken/wakened is also used in both transitive and intransitive senses.
I woke at four o’clock.
My wife woke me at four o’clock.
I was wakened at four o’clock.
In pre-20th-century texts wake may also have another meaning: to “stay awake”.
These verb uses are fairly complicated; but it is simplified by the fact that none of them are used much in colloquial English today (which is one reason why the past and participle forms are so variable). Instead, the phrasal verb wake up (past woke, past participle woken) is used in both transitive and intransitive senses:
I wake up at four most mornings.
My wife woke me up at seven o’clock.
He was woken up by a loud crash.
You will need to recognize the different forms and meanings in your reading; but for your own work you can use wake up in any context or register.
∗ marks an utterance as ungrammatical
As a native speaker of American English, I would most likely say
Yes, I do.
I could be a little more emphatic and say
Yes, I sure do.
If I wanted to use want I would most likely still use do:
Yes, I do want to.
But I would not say that often, and usually only when some person or some circumstance has cast some doubt on whether I want to.
The other option
Yes, I want to.
is not wrong, but it is probably the version I would least likely say. I may use it when I want to come but can't actually do so:
Yes, I want to, but I can't.
Instead of Yes, I do want to I could also say
Yes, I do do.
which means the same (emphasizing that I do actually want to go). However, this can be problematic, as do do sounds like doodoo, which is a slangy, child directed, or childish synonym of poopoo (excrement). So I wouldn't advise saying this unless you and the people you are talking with averse to any reference to this homonym, unless you wish to make a childish joke.
Best Answer
It's certainly true that "if yes" is a lot less common than "if so". But it's not so much that it's wrong, as that it implies things a bit differently and is therefore rarer.
From that ngram graph we can also see that, while putting "if so" in the middle of a sentence is almost as common as putting it at the beginning, "if yes" is very nearly always at the beginning. This points, again, to the difference in usage.
Basically, "if yes" only works if there's a question that is obviously answerable only with yes or no, and even there it's not necessarily preferred. In contrast, "if so" works even if there's no explicit question, or if it's a bit fuzzier. Examples where "if yes" is dubious or wrong:
Example where "if yes" is fine:
Unfortunately, the particular example you give is ungrammatical or unnatural in several ways:
"If yes" isn't actually wrong here, but it's not really ideal, since someone might have watched part of one, might be about to watch one, might have watched several, might have watched all of them…. So "yes"/"no" aren't exactly the only answers, and in this case, going with the fuzzier logic of "if so" works better.