Some (actually many) schools guarantee a certain number of years of funding for PhD students. For masters students, they are typically considered for funding only after the PhD students receive theirs.
Competitiveness absolutely depends on the specific program. Professional programs (e.g., MD/DO, DDS/DMD, PharmD, JD) are extremely competitive. Other grad programs vary significantly by school and by program, and no generalization can be made.
The number of grad schools people apply to depend on a lot of factors, such as their program of interest and what they are capable of affording. Applying to grad schools tends to be more expensive, however, people still typically apply to more grad schools than the number of undergrad schools they applied to. Those applying to medical school typically apply to 15 to 20 schools, sometimes up to 25. Those applying to dental school typically apply to roughly 10 schools. For others, I would imagine 10 schools sounds pretty reasonable.
Many grad programs do have graduation requirements outside of your major requirements, but they would typically be very few credits and make up only a very small portion of your degree requirements. The policy varies by school. In addition, for grad programs, you often will need an advisor to approve your coursework every semester and you get very little freedom in your choice of coursework, including those outside of your major requirements. You may take course outside of your degree if you want, but that will be in addition to what your advisor wants you to take for a particular term, thus you should only do so if you believe you can handle the coursework.
It’s not the norm for grad level courses to be in the evening, however some grad programs may be tailored to help those who need to work during the day. Not all grad students have day jobs, nor are they required to. However, if a grad student receives funding, they may receive it from working for the university as a TA or RA.