Learn English – Would you mind to answer this question

There is nothing wrong or weird about Happiest birthday. For example:

It was my happiest birthday as I received more presents than I expected.
Yesterday was the happiest birthday of my life.
I hope you will have the happiest birthday with your fiance in London.

As you mentioned, happiest is the superlative form of the adjective happy. It doesn't mean it is the most recent birthday. If your happiest birthday was 10 years ago, you could say, "10 years ago today was the happiest birthday of my life."

However, you don't usually say "Happiest birthday to you." when you wish someone a happy birthday.

"Would" is conditional. It can serve several purposes.

As an actual conditional, there would be a condition. "Would you agree...if...?" Your response could be a simple yes or no since the conditions have already been stipulated.

Similarly, if the question is "Do you agree...?", you could respond with a simple yes or no if the answer is that straightforward. If your answer is conditional, you could respond, "I would, if ..."

"Do you agree..." frames the question as a binary choice. "Would" can use its conditional nature to simply soften the question or ask it in a polite way. In that case, there is not an explicit condition, it is more a matter of affording some wiggle room in the answer by recognizing the possibility of conditions.

In this case, the response to "Would you agree..." could be yes or no, but it could also be the parallel construction "I would" or "I would not", without stating explicit conditions. That response would be ambiguous as to whether you actually mean yes/no, or the answer depends on conditions. You could also include explicit conditions.