This is an interesting question, and not one that people think about often enough. Understanding and being conscious of diversity is important for any staff member who will be encountering diverse groups of people, and those who work in nursing homes likely will encounter a wide variety. I think there are a few ways to deal with this, but the main answer is education. For a staff member to be more diversity-minded, they will need to be specifically trained to think in that way. Let's look at some examples for each key area you have.
Race, culture, and ethnicity: There are many ways a staff member could be trained to be more conscious of these factors. In terms of ethnicity, a staff member could be trained to speak or understand multiple languages. This would account for residents who may be more comfortable with a non-dominant language. Another important thing nurses should be trained upon is specific cultural differences. For instance, some Eastern cultures have very different conceptions of privacy than we do in the West. In this sense, staff should be taught specifically what these customs are, and should know to be mindful of them.
Disability: This is an important one, and awareness around it could also benefit from more intensive education. Teaching staffers about how to deal with disabilities, both physical and mental, would be advantageous in nursing facilities. Additionally, nurses and caregivers should be trained for emergency scenarios that can arise from different disabilities. An example of this would be learning how to help someone with epilepsy who is going through a seizure. The nursing home itself could also be provided with accessibility resources that can be used by patients. Ramps are important for mobility issues, and tools like sensory control devices may be helpful for people on the autism or ADHD spectrum.
Religious, transgender, generational: Similar to culture, I think the best way to deal with these issues is through education and training. Previously, there was a conception that all people should be treated exactly the same. I think this idea is starting to fade. People have key differences and deserve to have those differences honored and respected. One specific note: an easy way to help with transgender issues in nursing homes would be to install some gender neutral bathrooms.
Hopefully this helps! While there definitely are concrete resources that could be used in nursing homes, it is my opinion that the first line of defense is making sure staff members are respectful of and educated on people's differences.