Anglo-Saxon refers to the blending of Germanic and French roots. English is an Anglo-Saxon language because it is a mixture of Germanic- and Latin-root languages.
Not really. Anglo-Saxon is less recognized as a language family as it is a synonym for Old English. It is also an ethnic term for describing western Germanic tribes (Angles, Danes, Franks, Frisians, Jutes and Saxons) who came to colonize post-Roman southern Britain and people of their decent.
You are spot on about the Germanic and Latin roots. Back in the fifth century during the Anglo-Saxon migration, the intellectuals in southern Britain (and much of post-Roman Europe) spoke Latin. The spread of the Bible kept Latin as a influential language.
But Old English had only limited "French" influence. The Germanic Franks who lived in Gaul were never a major conquering force in Britain. There are some Old Frankish loanwords that influenced Old Saxon and Anglo-Frisian languages, but it wasn't much. The predominate French influence didn't come for centuries later via the Norman conquests. That resulted in Middle English, which is not synonymous for Anglo-Saxon.
According to the language experts, the classification is: English -> Anglic languages -> Anglo-Frisian languages -> Ingvaeonic languages -> West Germanic languages -> Germanic family. Ingvaeonic includes Old Saxon, but Anglo-Frisian does not. Likewise, West Germanic includes Old Frankish, but Ingvaeonic does not.