No startup funds for new tenure-track hire?
In my field (biological sciences, US), receiving no startup funding at all (at least for lab space and basic equipment) would be unusual. Maybe I could see a case where departmental overhead is sufficient to start a lab but I would still think of this as "startup money" even if established professors have access to the same pool.

However, asking funding agencies to pay your salary would be the usual course of business and the expectation by the funding agencies is that they would cover the portion of your salary corresponding to your effort towards the project: if the project is going to be about half of your work, it would pay at least half your salary.

If you have a gap in extramural funding, the expectation is that the institution would pick up your salary in the meantime, though this may come with other institutional responsibilities attached like an increased teaching load or some administrative work, especially if you have more than a brief interval without outside funding. I would be greatly worried if the institution isn't willing to contribute in this circumstance, but it is reasonable for them to ask for you to look outside for funding.

Funding for students in my field would also typically come from extramural grants: you would ask in a grant proposal that the work fund a graduate student to complete the experiments. Direct departmental support for students to do research is uncommon, but they may be supported by other external fellowships such as institutional training grants or grants directly to that student, or through a teaching or project assistantship through the department.

On a side note, I could see the situation for startup funding being a little different as an internal candidate, that is, if the institution is expecting you to continue working in the lab space you are in (i.e., if you are a postdoc or scientist, taking over some space from your current PI) and grow your independence over time.