Can iTunes sync podcasts to an iPhone without removing unfinished podcasts?
This IS actually the case. 


  What you should have a look at is the selection option that you have set. 


For instance, if you ask to only sync the "3 most recent unplayed episodes", it might be that the one you are listening to will be replace on your iPhone by the newly released episodes that have been downloaded in iTunes. When the episode you haven't finished will be one of the 3 candidates to be synced (i.e. "when you have listened to all the newer episodes"), the listening will continue where you were the last time you listen to it.

EDIT after the EDIT of the Question :

Hm, ok, with this edit my answer is not right anymore BUT i think you should have a way have it work as you want. If you select for instance "the oldest unread", the one you are listening to will be kept on the iPhone (if you listen to the episodes in order that they are realeased) Another trick is to use the checkboxes on the left of each episode and set the preference to "Sync only checked"