Even if the surface is frictionless, over time dust and random molecules will land on it due to its gravitational attraction. These won't sit still, they will slide around but remain on the surface at least. Over aeons the dust will form a layer and slow down its movement, eventually creating a more normal surface.
Space dust aside, if the surface was frictionless the rover would continue to move indefinitely without any energy added. Assuming the surface is perfectly flat, of course. If it is uneven the rover may become trapped somewhere, like falling into a hole (as would all the space dust...). It'd still continue to slide around in the bottom of the hole though.
It'll bounce on landing, depending on how rigid and elastic the surface and the rover are, but the vertical motion will settle down eventually as it would elsewhere, friction isn't involved. Any lateral motion will continue (see 1). Any rotational motion around the vertical axis will continue. If the planet is perfectly flat, the result of any landing will leave the rover gliding in one direction, rotating around it's vertical axis. You would need thrusters to dampen this motion, which can only be reduced, not eliminated entirely. A heavy rover with very fine thruster controls might be able to stay approximately still.
Yes, assuming the surface can be penetrated with the rovers tools, as long as they are pushed vertically into it then friction is irrelevant. Rotating drill bits would require thruster/reaction wheel stabilization, as there is nothing stopping the rover from counter-rotating.
The gravity on your planet is not high enough to hold an atmosphere.
Taking a sample on a perfectly flat, hard surface might be difficult. To be able to drill a core sample, the rover needs to hold itself stationary, and push against the surface with the drill. The first can be done with thrusters arranged around the sides of the rover. Pushing against the surface may be difficult though. The Philae lander tried to do this on a comet by using thrusters on the top of the lander, pointing into space, to push it downwards while drilling. In your case, the slightest vertical thrust which isn't exactly through the rovers center of mass will result in an efficient conversion of that thrust into the lander sliding sideways instead (much like pressing your foot at a non-vertical angle onto ice causes a skid). The side thrusters will need to be able to counter this. Any elasticity in the rover body will translate into an erratic skittering on the surface. The side jets need to be designed to handle this, too.
Finally, regarding drilling. I'm not sure how shear forces work in a drill bit, and if friction plays a role in getting the bit to "bite". Maybe frictionless means undrillable, this really needs further development...