For context, here is the song.

The most famous recording starts with Lehrer describing the Apollo missions as America "spending 20 billion dollars of your money to put some clown on the moon" and von Braun as a "good ol' American" before saying his name with a more traditional German pronunciation.

The song as a whole lays the destruction von Braun's weapons caused in London at his feet, paints him as completely apathetic to said destruction, and as being very mercenary about his political/national loyalties. Finally, it implies that von Braun is alreay planning a move to China.

My first question is how widespread were (what seem to be based on the song and its introduction) Lehrer's beliefs about von Braun being an unprincipled and disloyal foreigner who was brought to America solely because he would be useful to NASA (whose Apollo program Lehrer seems to critical of)?

My view of that era is that much of (especially white) America was supportive of NASA and that criticism of von Braun wasn't really a thing until decades later. The only criticism I know of was part of counter-culture like Gil Scot-Heron's Whitey on the Moon

Finally, to what extent was China's rise as a global power, capable of building its own space program that could draw in von Braun being foreseen at the time?