The three words are almost completely interchangeable.
I read your first couple of examples without realising you'd provided what you thought was the answer, and I didn't pick the same options you did!
For the first one (answers in a test) I'd lean towards "select".
For the second one (colours which look good on you) I'd use either "picked" or "chose" (NB past simple rather than past perfect... we very rarely use past perfect).
For numbers I agree with "pick"; this is just because there is a common phrase "pick a number from {x} to {y}"
For students who will attend the festival, I really can't choose which one would be best.
Thoughts on what differences there may be:
"Select" is generally only used where there are finite options. If you can only have red, green or blue, you could select one of them; but if you can have any RGB colour from 0-0-0 to 255-255-255 then "choose" or "pick" would likely be more appropriate.
"Select" also works well in the imperative, probably as it sounds slightly more formal than "choose" or "pick". Hence "Select from the following options" works well.
"Select" also works well with tickboxes (or checkboxes for Americans or IT people), whether they are on paper forms or on your computer screen.
"Pick" is quite informal. It works well with complaining: "Hurry up and pick one already!". It's also the most appropriate when you are physically taking an item, due to the (probably) related phrase "to pick something up".
"Choose", as per your definition, implies selecting what you want. e.g. "You can choose from hang-gliding, rock climbing or karting" works well; "Choose all the words that are about size" doesn't so much (that should be "Select").
All thoughts above are based on my experience as a native British English speaker.
If that is the actual quote (as you say) then first sentence is already not idiomatic English. It should be:
Reading your whole speech from your notes ...
"All of your speech" makes sense, but I just don't feel it's natural in this context. Also, for something like a speech or lecture, we write notes, plural, and read back from those notes. A "note" is something short, written for a particular purpose or to a particular person (or group).
None of the answer choices is really correct. The verb "involve" takes the prepositions "in" or "with".
I didn't want to get involved with them -- they're too radical for me.
I wanted to get more involved in my wife's family business, but her father wouldn't allow it.
"Involved to" is not idiomatic. Also, it's not really idiomatic to say the audience gets involved in a speaker. They can get involved in something like a speech or a movie, but not in a person (although you can get involved in someone's life).
Instead, in this context I would use a verb like "connect", or "relate".
Reading entirely from your notes is the least interesting way to deliver a speech. When you do this, you will find it more difficult to engage with your audience, and they will find it more difficult to relate to what you are saying.
I recommend finding a different book, preferably one written by native speakers.
Best Answer
From a strictly technical standpoint, both 2. and 4. are grammatically correct. An answer depends on the intention of the question.
But, having said that, I find a wide choice to be awkward because I'm not used to hearing it—I'm not even sure I'e ever heard it. (I'm used to a wide selection instead.)
If the word is choice, I think that large choice is more common. (Which is not an available option.)
In contrast, I've heard both wide variety and large variety. Don't ask me why; English is annoyingly illogical.
In conclusion, I would choose 2. because it is both grammatically correct and more common.