Is it permissible for me, as a dentist, to work in a mixed-gender environment due to family pressure, or to work with children without a niqab, given that it is officially prohibited by the government, knowing that I wish to help those in need?

The obligation for a woman to cover her face is a matter of scholarly disagreement. The view that the niqab is obligatory requires her not to unveil her face before non-mahram (definition) men and not to engage in unlawful mixing with men, except out of necessity. Working to earn a livelihood and help the poor is good, but it is conditional on not falling into something forbidden. Averting harm takes precedence over acquiring benefits. These benefits can be achieved through other means that are free from impermissible شرعي (Shar'i) aspects.