UPDATE: In version 10.0.9 Apple seem to have resolved most of the remaining issues with the use of Drop Zones.
You can now pick the first frame that you want to see in your DropZone clip by selecting it directly from the Browser.
Click on the Drop Zone image well, skim to the frame that you want, click and then Apply Clip.
You will now get consistent results even after export, and even after a restart (which could reset the chosen frame previously).
Note that you can longer use the older technique whereby you selected a clip directly off the timeline. Or rather you have to make a compound clip of it first and then select it. If you don't make a compound clip first you will get only a freeze of your source material.
The Drop Zone will use the compound clip from its start frame - you cannot choose a specific frame as you can if you select from the Browser. It might look as though you can when you hit Apply Clip but this is not the frame you will get.
I hope this information is correct - Drop Zones have never yet been reliable and theories as to how they work have frequently been subject to change. Here's hoping Apple have fixed the problem once and for all. Please do correct me if your experience differs from the above.
Further note for Motion Effect Creators: in some instances, Drop Zone functionality might be compromised because the original Motion Effect has not been made correctly. To ensure consistent speed behaviour within FCPX, make sure that the Motion Effect you are using has a locator positioned at its last frame and set to Project Loop End. (To set a marker, use Shift/M. To edit a marker, use Opt/Cmd/M. In the Type Selector dropdown, choose Project Loop End.)
I just discovered another way to use in & out points in drop zones.
Set a range (or at least an in-point) on a clip in the event library and mark as a favourite, add the favourite to a compound clip and add this to the drop zone. This allows for better media management and if you later want to adjust the in-point you can open the compound clip and do so. the drop zone won't update automatically in relation to the change but it can be reapplied.
Edit: I just had a hunch and checked something. You don't even need to set the favourite, just set a range and create a compound clip from that. Actually, that means favourites aren't the only way to do sub clips either, sub clips on steroids.