A lot of open show series do year end awards, but you do not get AMHR or AMHA points at open shows. The show organizer may give awards, though, at the end of the year.
The answer to your question would depend upon who's hosting the shows. Most shows award points based upon your placement in each class you enter. There might be a set value for 1st, 2nd, 3rd, etc. or a value that changes depending upon your placement and the number of horses in that class (as is done in AMHA shows). At the end of the show and/or season these points are usually tallied and High-Point awards given.
If you get your AMHR amateur information I believe you can have a judge or steward sign at the open shows you participate in for points towards your amateur status.