Does cosmology answer why the universe exist?
Not yet... but if it will find answer in the form like... Preuniverse state is highly symmetric and unstable, it must be crashing into a less stable state called universe... will you be satisfied?
What is the meaning of life in the eyes of a cosmologist?
The question about meaning of life is a tricky one since it potentially involves mental problem. If it is often, repeatedly, obsessively, appearing in mind it may be a symptom of a process of developing depression. The end of this stage is a conclusion life has no meaning thus no sense and the next stage starts with suicidal thoughts. Healthy way of approaching this question is to undertake serious study starting with literature survey and getting knowledge what answers others, professionals provided throughout history.