I would like to understand if skills/experience gained in rock climbing are going to be useful in climbing steep ice/hard-snow. Of course the gear as well as techniques involved in either are different, but is being a good rock climber helpful/useful in climbing ice and hard-snow on mountains?
There are some helpful basics like tieing into a rope and belaying that directly transfer from rock to ice climbing as they are the same regardless of the conditions.
Further transferable skills mainly depend on what kind of rock climbing you do. Sports climbing is a lot less helpful than trad climbing. Trusting your own protection is an essential skill in ice climbing and leading with double ropes can be very useful. A good head on run out climbs is useful as well.
To my experience, the physical side is only transferable to a small part as rock climbing is a lot about holding small holds while ice climbing is about holding your tools. These may seem like a jug but as you are holding them in the same position for long time you get pumped surprisingly fast. If course, good endurance is always helpful.
You are asking about ice and hard snow but these are actually quite different. On ice climbing (as in waterfall ice, typically steeper than 60 degrees) it is a lot about pulling and locking off. On steep snow (up to 55 to 60 degrees) it's more like walking as your power comes from the legs. Another difference is that snow can be hard to impossible to protect efficiently as opposed to ice which can be protected quite well with ice screws.
Tying in, ropework, belaying, commands and common sense transfer quite well from rock climbing. At least from trad. Someone with rock climbing experience could certainly top rope on ice or follow an experienced leader with only some brief instructions.
Ice climbing technique is quite different from (most) rock climbing. A rock climber is used to crimp small holds, palm slopers, handjam, and smear feet on slabs. All this is (virtually) absent on ice, where focus is on getting solid placements with your tools, locking off and stay balanced on your feet.
I've seen good rock climbers getting completely pumped om basic ice climbs and mediocre rock climbers cruising WI4-WI5 (granted not leading) on their first day on ice. So don't expect climbing skills to transfer.
Safety and the mental part
This is where it gets really different. Ice climbing is far more dangerous. Some handle this well, some freak out, others don't realize what risks they actually are taking.
To start with, while every experienced rock climber is aware of the risk of rock fall, that is not too common. On an ice climb there will be chunks of ice raining down constantly. Dont stand in the fall line while belaying! a big block of ice from 30m above is bad news.
Then there is the fact that ice occurs in winter. Ropes will freeze as will biners. And you. You will be wearing gloves. All in all, you will be clumsier. Tying knots and clipping ropes will take more time.
Ice screws are good protection if they are placed in solid ice. Problem is, solid ice is not always to be found when you need pro. Be prepared to deal with this before committing to a lead. When I lead on ice, I'm mentally soloing. My pro is a back up, there to keep me alive, not unharmed. Also, placing an ice screw is far more work than clipping a bolt or placing a cam. Placing an ice screw on steep ice, while hanging on to one tool is taxing. So much so that it might often be safer not to place pro and simply run it out until next good rest position. Therefore protection on ice is generally more sparse than on rock. On tho of this, falls are more dangerous, since you have sharp points everywhere that might hurt you in a fall, or your crampons might catch the ice giving you an instant broken ankle.
Oh, and beware of avalanche danger.
In short: while top roping on ice is not much more dangerous than on rock, things change dramatically once you lead. Some handle this well. Some like it (I do). But it's certainly different than rock climbs.