What do ultra-orthodox Jews believe the government should do?

Answer 1

Which government?

In Israel:

1) Many Jewish immigrants were encouraged or pressured (especially in the 1950s) to leave their Orthodoxy. This still happens, through such avenues as the media, which is largely very anti-Torah. Obviously, we religious Israelis would like to see that change.

2) The official, gov't sponsored majority school system (mamlakhti) is secular. Rates of alcohol-use, drug use and violence (for example) are much higher than among religious school students. Here too, we would like to see change. However, the secular establishment is so "allergic" to Torah that they prefer to endure the ills of their society (teen suicides, teen abortions, etc.), rather than try encouraging stronger Judaism as a solution.

3) The government has sponsored such forbidden things as digging up graves, public Sabbath-violations, and tens of other examples.

4) Unlike in the U.S.A., when a politician here (except in the minority religious parties) invokes mention of God, it creates a political brouhaha, as if a contemptible thing has been committed.

5) The religious community is discriminated against in housing, allocations, school funding, etc.

These are just a very few examples.

See also:

Orthodoxy and Zionism before the founding of modern Israel

Answer 2

Judaism teaches that Jews are to follow the law of the land unless said law attempts to force them to violate a religious prohibition. Murder for example.