Yes. Even overcast.

The only restriction for a private pilot and above is that the correct cloud clearances and visibility requirements for the airspace be maintained. Student pilots have additional restrictions, such as FAR 61.89(a)(7) which requires student pilots to always have "visual reference to the surface."

On the safety pilot issue, a safety pilot who is not acting as PIC generally only needs a private pilot certificate with category and class ratings and a medical certificate. But there is a potential "gotcha" in your situation - a safety pilot in a flight under IFR is required to have an instrument rating. That wasn't always the case. There was a change in the language of 61.55 in 2005. Safety pilots used to be exempt from 61.55 altogether. After the change (see 61.55(f)(4); if you have a pre-2006 FAR, compare that with 61.55(d)(4)), safety pilots are only exempt from the training requirements. There was a bit of a bru-ha-ha about the change since it wasn't highlighted (or even mentioned) in the Proposed Rule the way these things usually are.