OP, you said you want it done right. Doing it right will involve removing the baseboard and all the appliances. Using corner-round is primarily to cover up something (walls not square, hasty remodel, not laying out the new flooring to see where seams fall, etc.). If they remove the baseboard carefully you should be able to reuse it and caulk the seams/nail holes and then repaint. Another issue to consider is if you have any transitions from the new flooring to existing flooring. If the baseboard wraps around that transition the difference in flooring thickness can lead to other challenges (such as needing corner round to cover a gap if the new flooring is thinner, having to rip the baseboard to a different height if the new flooring is thicker, etc. I've seen in some homes where they have mitered the baseboard at the transition to account for this difference and personally I think it looks cheesy (but less expensive).
VillageTinker is right, you'll want to make sure you have the right clearance for your diswasher to slide in/out with the new flooring. 1/2" can make a big difference in some homes. Your installer should be able to measure for this with no problem. Removing the gas stove and the connecting line for the ice machine should not be a problem for a competent installer either. As with a lot of projects, a few extra hours of prep work will pay off in the long run.
Ask a lot of questions of the installer and make sure they know what you want and you have confidence they can do the quality of work that you are looking for.
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