What is the source for the idea that one should say 100 Berachoth in a day, and how serious of an obligation is this? Is it mandatory, is it laudatory, or is it merely a nice idea?
From the Tur (Orach Chaim 46): It is from a braisa based on pesukim. According to Natronai gaon, David haMelech set it into halacha to stop the deaths of 100 Jews daily where the spiritual cause of there death was unknown until "he delved and understood with ruach hakodesh and fixed" the 100 brachos.
So, it an obligation and is brought so in the Shulchan Aruch (Orach Chaim 46:3). Many have shown how if one davens appropriately, with the exception of shabbos, your 100 are covered.
From Menachos 43b
It was taught: R. Meir used to say, A man is bound to say one hundred blessings daily, as it is written, "וְעַתָּה, יִשְׂרָאֵל--מָה יְהוָה אֱלֹהֶיךָ, שֹׁאֵל מֵעִמָּךְ" - "And now, Israel, what does the Lord your God require of you?"
Rabbi Meir uses a play on the word מָה, saying instead - מאה. In effect, this then translates to - "And now, Israel, one hundred does the Lord your God require of you"
There are several reasons and allusions to this, listed here
According to this site, "the Chida (Machzik Bracha 290:1) writes that going out of the way to complete a hundred Brachot daily is only a virtuous practice. Sdei Chemed (Chet 34), and Sh"t Atret Paz 1:1 write that it’s a complete obligation. [Chazon Ovadyah (Shabbat vol 2 pg 339) writes that even the Chida may mean that it’s only virtuous to go out of the way to make the Bracha oneself but accumulating a hundred (including Brachot one listens to such as the Brachot of Torah reading) is an obligation]".
There is a machlokes on whether the obligation is Biblical or Rabbinic. See here for sources and details.