For alpine ice:
16 cm is plenty for both points of protection as well as the anchor assuming you take the time to clean off any surface crap. Nothing longer than an 18 unless it is a TR anchor for crevasse climbing then you want as long as you can get as screws will melt out during the day in the sun.
For water ice:
A selection of 10s, 13s, 16s and 18s should get you up most routes. Start with the longer lengths (16s and 18s) and as your skills improve start picking up the short lengths. Anchors will commonly depend on the quality/availability of ice but at least one 16 and an 18 is great.
I have some used Grivel screws for sale... shoot me a PM
EDIT: this only applies to stainless steel screws. Ti or anything else that has smaller threads needs more length