I just recieved my diplomma (actually a sort of certificate) for my 6. kyu about two hours ago, and i still have that feeling. I bowed out of true respect and thankfulness to my sensei. I thanked him mentally for teaching him, and it manifested itself as a bow. It's really a great feeling when someone helps you, or teaches you, and by the smile on their face, you can see that it was a pleasure for them too. In that moment, my ego was gone.
I know that some people at my dojo 'fake' bows, and that they bow just out of habit. Maybe i am wrong, but that's just my opinion. You can't see they are giving respect, but all you see is some bow-like movement of the upper part of the body. They are bowing with their bodies, and not with their minds. Sure, that has happened to me many times, when i am thinking of something else than aikido when i am entering the dojo
But upon leaving the tatami, i am really glad that it is there, as i know how it is to practice without it (especially for a beginner). I bow to the dojo, because it is giving us a place to train, it's warm inside, and we are protected from the elements. I also bow to my sensei when i feel appropriate during training. It is a sort of tradition to bow every time when sensei shows you something in your technique, but there are times when i really feel thankful for that. That's when i truly bow.
For example, we were doing uchi kaiten sankyo. There was a minor mistake in the way i took grip of uke's hand, and that rendered the whole technique useless. Uke would fall, but it would be fake, and i could feel it. I asked my sensei about the issue, and he was uke for me. He felt what i did wrong and corrected me. It was just a small misalignment of my thumb. But i was very thankful for helping me, because it removed a huge obstacle out of my way. Without it, i would be stuck at sankyo for a long time. But he took the effort, and solved my problem in just a few seconds.
I see bowing as showing respect to anything you can think of. I bowed when going to a friend's dojo as an observer. Well, if they had trainings in the open, and it was raining, i would have to watch the training soaking wet... It makes sense. Of course, i did not bow to the tatami. I have respect for it, but it would be faking, because i can't be truly thankful for it's exsitance.
Now this was my longest post ever...
I hope i wasn't confusing, and please forgive me if i was, since english is not my native language. Our common language is that of Aikido. Let us train it to proficiency.