Is program analysis part of programming language theory?
"Programming language theory" more or less includes everything you have listed, but of course there is overlap with other areas, such as software engineering, algorithms, security, human-computer interaction, and good old mathematics.

For instance, if you look at last year's programme of the Principles of Programming Languages (POPL) conference you will see not only topics about design, implementation and semantics of programming languages, but also about security, program analysis, papers about resource complexity, and many others.

In general I would say it good to be open about what "belongs" to a "research area". It is a shame to miss opportunities for new discoveries because of narrow-mindedness.