As noted in the question, Heinlein's writing career was long and started a very long time ago (1939-1987). This is a man who was born in the Bible Belt in 1907 and spent the first half of his life vigorously interrogating his own upbringing and coming to radically different conclusions about ethics and religion than the conventional wisdom of his native time and place. He had a considerable stylistic range and wrote on many different themes for some very different audiences.
As background, Heinlein was a socialist in the 30's, while later in life he tended to express libertarian views. His wife Ginny was a biochemist and a navy (WAVES) veteran. There is a story that when he was used on TV as a color commentator during the Apollo program, he flustered Walter Cronkite by suggesting that women should be allowed to be astronauts. However, his views with respect to feminism and women's roles were certainly not PC by modern standards.
Early career, 1939-1947
Heinlein's fiction from this period shows him still working on putting together his style, with some of his work such as For Us, The Living and Rocket Ship Galileo being noticeably below his later level of proficiency. This period provides the best ammunition for those who want to portray Heinlein as racist and misogynistic.
In 1941, he published two novellas in Astounding, "Universe" and "Common Sense," which were later combined into a fix-up novel called Orphans of the Sky. They depict a society aboard a generation starship that has degenerated into ignorance, superstition, cannibalism, and slavery. Slavery includes all or nearly all women, who are treated as chattels, are named by their men, and are subject to casual physical abuse. (There is one female character, a knife maker, who lives independently and is afforded some respect because her craft is important, but she is beaten down by one of the sympathetic male lead characters when she attempts to stand up to him.) Whereas the other negative aspects of this society are treated very creatively, from multiple angles, and in considerable depth (e.g., a sympathetic character narrowly avoids being eaten), this is not so for its misgyny. Author Amal El-Mohtar writes that the book is:
often interesting, and competently written, except for the bit where every aspect of this society's ignorance is complicated and problematized and addressed -- except for the women-are-silent-chattel aspect, which up until the very last page of the novel, is taken as read.
If anything, I think she understates the problem with this particular work. Reading it, I get a disturbing feeling that violent subjugation of women is supposed to be a joke, for the amusement of male readers.
Orphans of the Sky does, however, seem to me to be completely atypical of Heinlein's treatment of women, even in this early period, when he was already showing considerable creativity in his treatment of gender roles. For example, there is a scene in the 1942 Beyond This Horizon in which two male characters compare notes on their badass firearms, after which one of them compliments the other on the shade of his nail polish.
Maturing style, 1947-1958
Toward the end of the 1940's, Heinlein broke away from the stereotyped pulp fiction style and became the first science fiction author to sell stories to the "slick" magazines such as the Saturday Evening Post. A big chunk of his writing during this period was his juvenile novels, many of which were serialized in the Boy Scouts magazine Boy's Life before being published as books. These are written for a nominal audience of teenage boys (although Heinlein prided himself on not talking down to his audience, and fought a running battle with his prudish editor over their contents). In some of these, girls and women are simply absent. One (Podkayne of Mars) has a female lead who is somewhat saintly and has a younger brother who [spoiler 1]
In several of the juveniles, there is the theme of the boy protagonist who is forthright and courageous but not terribly bright, and has to be helped by a more intelligent and sophisticated girl (The Star Beast, Have Space Suit--Will Travel). In Starship Troopers (originally submitted to his publishers as a juvenile), the protagonist ends up as a grunt in the interstellar marines because he isn't as good at math and electronics as his high school friends, but one of the women his age becomes a starship pilot. In two of the juveniles [spoiler 2] there is a surprise revelation that a character thought to be one sex is actually the another.
One of the juvenile novels (The Rolling Stones) features a cantankerous, hypercompetent grandmother who is an experienced spaceship pilot. In Starman Jones the teenage female lead hides her intelligence until it is revealed later, and in The Star Beast she sexually manipulates an older man (or tries to?) to humorous effect.
Mature style, 1958-1969
In Heinlein's books for mature audiences from before 1970, we see some of the same shticks, such as the confused or less intelligent man paired with the more experienced and intelligent woman (Glory Road, For Us The Living), and the hypercompetent woman (Glory Road, Barbara in Farnham's Freehold). In Stranger in a Strange Land, the protagonist manipulates his own sexual biology to give himself an androgynous body. In another of his works from this period, when he was at the peak of his powers, the computer in The Moon is a Harsh Mistress consciously switches back and forth between being psychologically male and female.
Later in life, Heinlein ran into a series of health challenges that he feared had clouded his thinking, and he began to doubt his own ability as a writer. Around this time, he stopped accepting requests for revisions from editors, and started getting feedback mainly from his wife, Ginny. My judgment is that his writing from this period is almost all awful, but others disagree.
One of the standard critical complaints about his novels from this period is that the characters are one-dimensional and not believable, and I think this applied to both his men and his women. Cases in point are the sexy madonnas of I Will Fear No Evil and Time Enough for Love. Even more cringeworthy is Deety in The Number of the Beast, who spends a lot of time thinking about how happy she is that her husband likes her big breasts. Friday, one of his stronger works from this period, has a female protagonist who is an artificial person and disbelieves her own self-worth, although the narratives show her to be a superior person in various ways. To Sail Beyond the Sunset features a female POV who is filled in from her sexy madonna depiction in earlier books.
Another common criticism of this period is that Heinlein often uses characters as mouthpieces for his own opinions. This is more frequently done with male mouthpieces, but there are female examples (Maureen in To Sail Beyond the Sunset, two of the POVs in The Number of the Beast). (Something similar, although less overt, happens in the earlier Glory Road, where the female lead Star voices Heinlein's political opinions.)
One thing to keep in mind while putting all of this in perspective is that during most of Heinlein's early writing career, he was unusual among SF writers simply for writing fleshed-out characters at all. He made much more sophisticated use of voice and point of view than his peers in the "big three" Clarke and Asimov.
smuggles drugs and get hold of a nuclear weapon.
In Tunnel in the Sky, a minor boy character is later revealed to have been a girl who pretended to be a boy. In The Star Beast, the alien (who is actually the star of the show) is originally referred to as male, but is later found to be a kind of queen or princess (although her race has more than one sex).