Gosh am I glad you posted. Your Joshua Adams shows up in early records around my Clay and Jackson Co Adams over and over. He has driven me crazy trying to find how he fits in... That list of our Adams siblings comes from a 1835 Will settlement, so it only tells us who was still alive in 1835.
Your 1775-76 birth for Joshua Adams along with his father being Alfred Adams takes a load off my mind. The parents of all our Missouri Adams clan were married in 1786.I went through all our Albemarle and Fluvanna Adams family lists and never found an Alfred or Joshua. Those Virginia families used the same names over and over, so I'm pretty comfortable in saying your Joshua is not directly related to our clan.
Adams research is so tricky, and there are Adams absolutely everywhere. I can recommend the Adams Addenda microfiche from Salt Lake FHL, it covers the 1970-1990 issues. I found lots of useful clues and a great 15 page research paper on our Adams. FHL # 6047027(42 fiche @ 15 cents each)
As for California, I know there were some adventurers who were good friends of our Jackson Co Adams clan. But I really don't know much about their trips. I was not aware that ourLynchburg was ever in Calif.
Afraid I am no real help on your Joshua and Alfred Adams.