What’s the reference re: “ice rocket”? Google search turned up Losers in Space.

Best bet I got is a reference to the book "Seveneves" which uses a nuclear reactor buried in primarily ice asteroid. They consume the ice and use steam essentially as a propellant.

Basically, a rocket that's 100% propellant, no onboard energy. Use the laser to a melt the ice and create a cometary-like exhaust trail, e.g. a purely ablative rocket.

Since the pulse only lasts for a few femtoseconds, no.