I can't answer for the California region specific, as that's outside my own area of expertise. Hopefully someone more knowledge on that front will be along shortly. In the meantime, though, I thought I might tackle some of your questions that have more general application.

How are LGBTQ peoples treated in Native American communities? I heard the phrase two-spirited: does this mean that Californian Native Americans consider these people to be both male and female?

"Two-Spirit" and "Two-Spirited" is a modern term that was developed and adopted by a portion of the Native gay and lesbian community in the early 1990s (with some possible precedent developing in the late 1980s). The term is a literal translation of the Ojibwe phrase niizh manitoag, which seems to be its origins. This was still a neologism, as far as I'm aware. The historical Ojibwe names seems for male and female alternate genders appears to have been agokwa and okitcitakwe, respectively (the latter definitely seems to have come down to us from a time before the modern Ojibwe orthography developed--"okitcita" would now be spelled "ogichidaa"). It's since be used, occasionally, as a catch-all anthropological term for the myriad varieties of non-binary genders found in various Native societies, replacing the older, and often pejorative "berdache." In many (if not most) Native societies, alternate-gender people were often viewed as liminal entities--occupying two seemingly mutually exclusive categories simultaneous--bridge the world of men and women and potentially being able to access the power of both (or perhaps gaining access to new power unique to the alternate-gender's own role in society). They didn't fit neatly into male or female categories, being a bit of both. The term "Two-Spirit" was adopted to revive and reimagine the old traditions in a modern context.

As for the Californian nations, yes, many of them did have alternate genders in addition to male and female. Hopefully someone else can tell you about them though.

Is it true that roughly 2/3 of Native American communities were matriarchal?

No, far from it. Women held considerable political power in some nations (like the Haudenosaunee, as the premier example) and potentially little in others. But not even among the Haudenosaunee would I say that women held such exclusive power to call their society matriarchal.

Now, perhaps you meant matrilineal instead--that is, calculating descent through the female line. That was / is widespread, particularly in the part of the Americas I focus on. I know it's somewhat less common on the Plains, and further west (such as in California), I couldn't say.

What meaning does the (English) phrase "medicine" or "good medicine" mean in Native American culture? The speaker used these words a lot. Does this have a direct translation in different Native languages in California? How far does its usage date back?

Medicine in a Native context covers a fairly broad range of topics, involving a mix of medical and spiritual practices and knowledge from a host of cultures spanning the Americas. Medicine involves the use of herbal remedies and religious rituals and objects to promote the health and success of individuals and communities. Though there are some commonalities, the specific remedies, rituals, and objects used in medicine will vary from nation to nation. But in the wrong hands, the skills involved with medicine can be used to inflict harm (either intentionally or accidentally), resulting in what in English is often called "bad medicine," "poisoning," and "witchcraft."

May I suggest the Southwest History Museum as a good source of information? If nothing else, they have a very good library.

I don't have specific answers to all your questions, but I can recommend some authors who have written a lot on California Indians. Steve Hackel's Children of Coyote, Missionaries of St. Francis examines Indian-Spanish relations from 1769 to 1850, and while it is not really an ethnography of various California groups, it would deal with many of the issues you raise above. Doug Monroy's Thrown Among Strangers is an older account of similar processes, and could be useful, as might Lizabeth Haas's Conquests and Historical Identities in California.

These are really historical accounts though, written by historians who privilege written sources, and thus dealing with Indian-Spanish, -Mexican, and -Anglo-American relations as opposed to California Indians on their own. What you probably want are ethnographies of California Indian groups, so perhaps /r/AskAnthropology would be better able to help there. I know there are many ethnographies from the early 20th century, but you should approach those with caution given their historical situation. I have no idea what modern scholarship on California Indians looks like.

Approximately how many groups were there pre-contact? How many are there now? How many of these California groups are federally recognized? Is it a sore point amongst Californian Native Americans over who has federal recognition and who does not?

Estimates vary somewhat depending on what might be considered a "group". There were probably upwards of 500 politically autonomous "groups" in pre-contact California. The Federal government recognizes slightly more than 100 tribes in modern times, with another 80 or so not having it. These unrecognized groups are still assigned to tribes, but they want independent recognition. This is somewhat of a sore point for certain groups. The San Gabriel tribe comes to mind, with multiple internal factions seeking independence from the federally recognized group.

What is the state government of California's track record for relations with the Native Americans? Is it true that the foster care system has damaged Native American communities?

California-Native relations have generally not been spectacularly good for the Natives. Forced boarding and curriculum BIA schools had an enormous effect on native culture. The foster care system was little better. For raw numbers, more than 20% of native minors were living in a non-indian foster homes around 1971.^2 Criticism of the foster system has been widespread.

To what degree to Native Americans in California have a shared culture as opposed to regional affiliations?

To a very significant degree, they do not have a shared history or ancestry. There are more than 100 distinct languages recognized in California today, with several hundred dialects recognized beyond that. In my area of expertise, the Southwest, it's not uncommon to talk about the Patayan and other desert peoples. To the best of our knowledge, they generally had more in common culturally than with the Pacific coast tribes. The "Pacific coast tribes" themselves were very distinct, with several hundred known and many speaking entirely distinct languages. Settlement of California by the Americans led to relatively even-handed suppression of everyone. To some extent, the various tribes can and do treat that as a uniting factor, alongside modern ideals and history.

^1 Pritzker, Barry M. (2000). A Native American Encyclopedia: History, Culture, and Peoples. pp. 112, Oxford: Oxford University Press.

^2 Indian Child Welfare Act of 1978: A Response to the Threat to Indian Culture Caused by Foster and Adoptive Placements of Indian Children Manuel P. Guerrero American Indian Law Review, Vol. 7, No. 1 (1979), pp. 51-77, University of Oklahoma College of Law

Two resources that will help with your questions are The Handbook of California Indians by Alfred Kroeber, 1970 and the Smithsonian's Handbook of North American Indians, California Volume edited by Robert Heizer. Others have answered many of your questions. I will take on only one.

Is it a sore point amongst Californian Native Americans over who has federal recognition and who does not?

It is an extremely sore point because there was no rhyme or reason for who were recognized and who were not. Many are under the false impression that tribes became recognized in California by way of treaties. Although many tribes signed treaties, which they never understood, and these treaties were signed and agreed upon by federal Indian agents, no treaties with California Indians were ever ratified by the senate. As Heizer points out (Smithsonian's Handbook, pg. 704), "Legally speaking, no California Indian tribe (among those considered in this volume) ever entered into relations based on treaty with the U.S. Government." Originally congress established, by law, seven "military reservations" where natives we're provided with housing, schools and other support. However, in order to gain these benefits you had to move onto the reservations. Most natives not forced to move were unwilling to do so. Subsequently, more reservations have been established and tribes recognized. However, there is absolutely no comparison between the anthropologically identified tribes and federal recognition. The process for recognition is still active but the burdens are onerous and the average waiting time for consideration is 15 years. So there are many native California Indians who still follow traditional practices, some who still speak their native tongues who had their land and gathering rights stolen and received no compensation or recognition.