It's not the feeling of anxiety [Angst] itself that is important as much as what it reveals to Dasein. As Hubert Dreyfus put it:
"The anxious Dasein can still see that there is a whole system of roles and equipment that can be used by anyone, but, just for that very reason, this system has no essential relation to it. Equipment is still present with its in-order-tos, but Dasein no longer experiences itself as assigned to a for-the-sake-of-which and so lacks a reason for using it"
Dreyfus, H, 1991, Being-in-the-world, The MIT Press, Massachussetts (p.180)
Through anxiety, the systems in place that have been inherited through cultural transmission lose their seemingly inherent meaning. The comfortable, guiding structures fall away and Dasein can see itself in a certain kind of nakedness.
As Heidegger himself wrote:
“Initially and for the most part, Dasein does not have any explicit or even theoretical knowledge of the fact that it is delivered over to its death … Throwness into death reveals itself to it more primordially and penetratingly in the attunement of anxiety”
Heidegger, M, 2010, Being and time, State University of New York Press, Albany (p.241 [§50])
The more general, everyday feeling of anxiety about something (a test or a relationship, etc.) is not something that is originally abstracted upon to then create a larger cosmological picture. Contrary to that conception, Dasein understands this everyday feeling of anxiety because of the underlying being of its own being (to use Heidegger's expression). In other words, the constitution of Dasein allows for the everyday feeling of anxiety.
Happiness is a contented feeling, and not something that would typically bring Dasein face-to-face with the underlying abyss of its being. Again to borrow from Dreyfus:
"Just as the breakdown of a piece of equipment reveals the nature both of equipmentality and of the referential whole, so anxiety serves as a breakdown that reveals the nature of Dasein and its world ... In anxiety, inauthentic Dasein experiences the world as an instrument that has failed to do its job"
Dreyfus, H, 1991, Being-in-the-world, The MIT Press, Massachussetts (p.177-9)
Death in relation to Dasein is a confrontation with Dasein's own finitude. Anxiety reveals the way in which Dasein is individualised, it shows how it is made singular from the rest of world of human beings (i.e., from the They). This revelation allows Dasein to take an authentic stance on its own existence rather than to simply fall prey to the public world. According to Heidegger, the idea of subjectivity is taken from this more primary ontological existence, and it is an error to look for grounding in any kind of objective/subjective position. Anxiety is prior to subjectivity because it is an ontological feature that allows for individualisation in the first place.