Fuzzy about mipmaps and LODs- questions and assumptions

Could someone clarify the following?

My assumption is that for mipmaps to be generated one of 2 following conditions must be met

1- Either your texture must be a power of 2 size or ‘Power of Two’ must be checked in the Texture Map settings.

2- With mipmaps enabled some geometry very close to camera has low res or blurry textures. Why is this happening? Shouldn’t the proper mipmap texture automatically be used?

3- What is the connection between LOD’s which I assume are mesh specific, and mipmaps?

Thanks for any feedback!