When you want to say cute, for example, 很可爱 means cute without very, whereas 非常可爱 actually means very cute

I have the same question. Also, which of these expressions is stronger?

非常=extremely

Yes, I'd have used the latter

Order of magnitude. 非常 Is higher than 好 in magnitude.

Typically in Chinese grammar, time is always placed before the subject. So when asking about weekends for example: "nǐ zhōu mò xǐ huān shén me?" (You like to do what on the weekends?). The time goes before the question or (you, weekends, like to do what?) I like to think of it as kind of asking about "your weekends".

我们今天非常高兴. Yes.

Yes, 我们今天非常高兴 is okay.

Yep! Either directly before, or directly after the subject.

Yes, just do it and the answer is accepted.

My mandarin teacher says its correct either way

Good question! Let's isolate the main grammar point here, which is this: why is 是 not used in this kind of sentence? 是 is not used when there is only a pronoun and an adjective without a noun equating the pronoun with something:

我很高兴。 你很高兴。 他很高兴。

Other adjectives may be used, of course:

我很害怕。 你很聪明。 他很难看。

You may use another adverb instead of 很 if you want:

我特别害怕。 你特别聪明。 他特别难看。

So, back to the question of using 是 ... if you are saying something that essentially equates the (subject) pronoun with that noun, then you use 是, whether or not you have adjectives:

我是美国人。 你是学生。 他是老师。

Again, you may use adverbs and adjectives if you want:

我是很自信的人。 你是很用功的学生。 他是很好的老师。

In your sentence, 是 most likely works as 确实 què shí (indeed) or 确实是. i.e. it may be the same as 今天我们确实(是)很高兴. This partly explains why we normally use 很 instead of 是 before the adjectives as in 我们很高兴. Here is a gross summary:

• 是 + adjective: (slightly) emphasizes the attribute/state
• 是 很 + adjective: ditto
• 很 + adjective: just means being in a very state
• 很 是 + adjective: ditto, but tends to mean “very”

Examples (the context matters):

• How do I feel?
  • 我 很高兴 — I am very happy.
  • 我 很是高兴 — I am very happy.
  • 我 不高兴 — I am not happy.
  • 我 不很高兴 — I am not very happy.
  • 我 不是很高兴 — I am not very happy.
• Am I not happy?
  • 我 是 很高兴 — I am indeed very happy.
  • 我 是 高兴 — I am indeed happy.
  • 我 不是 高兴 — Actually I am not (being) happy, but rather …
  • 我 不是 很高兴 — Actually I am not (being) very happy, but rather …
• How are you (two)? — 是 adds some stress for comparison
  • 我(是)高兴，她(是)不高兴 — I am happy. She is unhappy.

P.S. It is better to always put a stress on 是, as in 我是(很)高兴, in order to make “indeed” more prominent.

Update: Generally I think 很 means “very/really/quite”.

would be correct

We can say that in a very casual manner, likely that we just feel like mentioning 今天 at the last moment but it is just a small complement and not very important. Usually we put the time at the start of the sentence or before a predicate which can be 非常高兴, i.e. we can also say 我们今天非常高兴.

It's okay in colloquial context. In casual talks, ''好''=''非常''≈''很'' Often we put interjections like '啊' or '呀' when we use '好' as 'very'.(It's okay if you don't, it's only customary) For example, 我今天好高兴啊（呀）=我今天非常高兴 and is slightly stronger than 我今天很高兴. Hope it helps:)