I can readily understand if it means "much more", not just "more" since たくさん means "much". Does it imply it? If it doesn't, what if one wants a little more only not much more? How should one say it?
Just たくさん means "much/many", but もっとたくさん doesn't show the difference is "much/many". You need to notice that たくさん is used in comparison there. For instance, even in a situation where 3 is not たくさん(= many), 3 is もっとたくさん than 2. Similarly, even when 5 is not high, 5 is higher(もっと高い) than 4, or even when 1000 is not few, 1000 is fewer(もっと少ない) than 1020. Thus, もっとたくさん just means "more".
I already showed how to say "a little more" in my previous post, by the way...
Are you saying もっとしたいです is still okay?
In that context, yes.
Could they actually mean the following?
少し大きすぎる/小さすぎる
少し高すぎる/
少し多すぎる/少なすぎる
~すぎる expresses "excessive", i.e., "too", so 少し大きすぎる is not the same as 少し大きい in nuance or connotation. The same goes to other examples.
Both もう少し and もう少したくさん mean the same which is a little more and たくさん in the latter can be redundant. Do I understand correctly?
In your textbook's context, yes, もう少し and もう少したくさん are the same in meaning, but these two words are not always the same in meaning. For instance, just もっと or もう少し can be used also for the time length, as the same meaning as もっと長く or もう少し長く. Think about the following situation.
日本語のクラスは授業が短いと思いませんか。
Don't you think lessons are too short in Japanese class?
そうですね。もっと長くしたいですね。
I agree. We want to do more.
The second person's answer can be もっとしたいですね。 since it's obvious they are talking about the time length of lessons from the context. However, note that there can be a situation where it's ambiguous which the speaker is talking about, quantity or time length (or other possibilities, e.g., the number of the lessons もっと(何度も)したい, are also possible). たくさん (or 長く, 何度も ) is used to make the meaning more clearly, as I wrote.
Are both below correct grammatically?
もう少しください
もう少したくさんください
Yes.
Apart from above, もう can only be used in the following examples or the like to mean more. Am I correct?
もう一度
もう一回
もう二回
もう or "もう + interrogative + も" can mean "no more/not anymore" in negative sentences.
e.g.
もう誰も来ない。
No one will come anymore.
もういらない。
I don't need anymore.
もう食べるな。
Do not eat anymore.