Is there a proper and respectful means of disposal of old kippot? I have lots lying around. I set aside many of them that are dingy and aren't ever going to be worn again by anyone. What can I do with them?
user218076's answer is correct. But, as you also asked "What can I do with them?" you could donate them to a shul, esp. a Conservative or Reform shul. Depending on the kippa style, they may want some or all of them.
Another good donation place is a yeshiva elementary school. While this may sound counter-intuitive, little kids have a habit of losing, misplacing them or just plain not bothering to show up to school wearing one. Yeshiva rebbes are extremely insistent on kids wearing one at all times. So, the principal's office and rebbe's should have an ample spare supply. Believe me, when I was a kid, one time they sent me home all because I forgot to wear my kippah. It took my grandma to yell at the principal to keep a supply in his drawer. So, if you don't mind, you can "salvage" my youth horror story :-)
Religious summer camps have a similar problem as above, but in higher volume, because there are more sports (a game like basketball tends to create kippa fly-aways!) and camp bunks create a higher likelihood of lost kippot (Sad to say, but even Jewish religious kids, esp. in camp, steal stuff.)
A few communities such as mine have a religious online post forum where people post things they sell, donate or want. (Example: firstname.lastname@example.org) If your community has such a forum, post that you are donating these. Last time I had a bag of about 50 kippot. I posted, and I was amazed how quickly various people came to my home to get some of them.