How do I become motivated after unfairness in research?
In both of the experiences you describe, you were not treated fairly.  I think it's really healthy to be disappointed at unfair treatment: for one thing, this kind of disappointment is a key component in learning to treat others fairly; for another, indifference to being treated poorly could be dangerous just as absence of pain on being burned could be dangerous.

It is however good to cultivate a sense of perspective on such matters.  When unfair treatment has serious, lasting negative consequences, you should fight back.  When it's a passing thing, sometimes it is better to take just one quick shot at rectifying the situation, and sometimes it is better just to let it go.  Let me ask you this: What are the specific negative consequences of what happened to you?  Are they durable?  Will they affect your future?


With regard to the first case: I am a tenured full professor, and I have never had my academic research formally judged and graded at a conference.  So I honestly don't know what you lost out on by being judged poorly.  Maybe some words of praise, a line on your CV, and/or a small cash prize?  These are not serious losses.  The experience seems to have hurt your morale.  I understand that, but I think you can easily shake it off with the right perspective.  Two people evaluated your research.  One person gave you an excellent evaluation, and one person gave you a poor evaluation.  Already that's not so bad.  But wait, it gets much better: the person who evaluated your work poorly had you confused with someone else.  So actually no one really thought your work was poor.  Isn't that great?
The second situation is similar in that in fact you are only receiving positive feedback.  If going to a conference in Pittsburgh is praiseworthy and you went to the conference, then the praise actually applies to you.  The person who gave the praise may not know that, but you know it, and that's much more important.  By the way, when a student award is for something subjective like "good research" rather than being best with respect to an explicit metric, awards are very often given to graduating students rather than non-graduating students.  If an award was given to a graduating student in your research group and you feel more accomplished already than that student, you have quite a good shot at getting the award yourself when you graduate.


There is a general theme here: successful academics are largely self-motivated.  And for most academics, the greatest reward and job perk is the satisfaction they derive from their own understanding and achievements, both at any given time and the experience of being on a steady upward trajectory of mastery and accomplishments.  (I am reminded of my college's alma mater: Let knowledge grow from more to more; and so be human life enriched.)  Praise and rewards are not insignificant -- among other things, they are key checks on our own self-evaluation -- but they ought to be less significant than your internal sense of your own academic trajectory.  If you actually love research -- and it sounds like you do -- then that is your reward right there.  I'm not saying you should ignore the rest, but as above you can try to put it in perspective.

To address your questions: 


  Is this kind of thing common in academia?


The specific nature of the bumbling you describe is not necessarily so common (though bumbling of one kind or another is hardly rare).  But being underappreciated largely by mistake or inattention is rather common, I'm afraid.  Every academic I know can spin a wrenching yarn about excellent work which was treated rather shabbily by others.  


  More importantly, how do I feel confident and motivated again in research when I feel really unsupported?


That's a very personal question, but I have given some part of what I hope is a helpful answer.  Namely, you have to do you.  Being unsupported through what you can clearly and rationally perceive are the faults of others rather than your own should not discourage you from continuing in academia.  It may mean that there is a better place for you, but good news -- repeated changes of location which bring new mentors, collaborators and subsidiaries are a key feature of academia.  As I mentioned, in each of your stories above there is clear positive feedback on your work and no clear negative feedback -- so I see absolutely no reason for you to lose confidence in yourself and in your own work.  

The point of being in an "honors college" is not accumulating honors, by the way.  (To be frank, the terminology is really pretty silly, and even somewhat off-putting.)  Rather it is about being given extra opportunities and being guided towards more rapid mastery: to say it slightly mathematically, it is about increasing the slope of your academic trajectory.  Giving a student an award for going to a conference is also a bit silly, like giving a child an award for bringing home a good book from the library: yes, there was some praiseworthy behavior involved, but the important part is the opportunity, not the praise for receiving it, and whether the opportunity was used well and/or turns out to be significant is for the person who received the opportunity to judge, not the person who gave the praise.  Would you rather have the praise and not the book or the book and not the praise?  Clearly the book is preferable...if you actually want to read it.  Similarly, I hope you will see that by going to the conference and not getting the praise, you were supported more (in this one instance, anyway) than your colleague who was lavishly praised for a conference that she didn't attend!  Ask yourself whether you enjoyed the conference.  If you did, there's your motivation.