Gustav II Adolf (as he's called in Sweden), Gustavus Adolphus is the latin version of his name, did use propaganda a lot, and it seems like him and his administration were pretty good at it. It can thus be pretty hard to pick out what he wanted people to believe about him and his family life and what things actually were. So anything you read about the subject can and will be mired in the propaganda of the time.

That said, it seems like Gustav Adolf, despite fathering bastards, were at least a passable family man. These seem to have been genuine love between him and the queen, at least from her side and she mourned him to levels considered unhealthy and even hysterical by riksrådet (the realm high council), which separated her from Kristina, fearing she would affect the young girl in a negative way. The king and queen had suffered the death of three small children (two stillborn, one that did not survive infancy) before Kristina was born, and while the king was probably hoping for a son, he was still elated about having a healthy child, regardless of the sex.

Kristina seem to have loved and admired her father, and he seem to have doted on and loved his daughter, but also made sure that she would get a stellar education to be able to become a regent should he not have a son. There are stories of the king having Kristina ride on his shoulders as he inspected the fleet before sailing for Germany in 1630, pointing out ships and telling her stories of them and teaching her how to make out how powerful a warship was in speed, armament and so on.

On the subject of Gustav Adolf's bastard, Gustaf Gustafsson af Wasaborg, he was born 4 years before Gustav Adolf's marriage to Maria Eleonora, and it seems like few considered a single bastard, especially one born before marriage, a problematic situation. As he was born outside wedlock, he was not in line for the throne. While the Swedish parliament held the right to choose heirs for the throne if the line of succession was not immediate - there were both Karl Gustav, her cousin (a successful general, a potential suitor and a trueborn), and the Polish Vasas (if anyone could be convinced to convert to protestantism) in line fore any kind of bastard could be elected. Most powerful men of the era produced many bastards, and Gustav Adolf seem to have put himself under more restraint than most rulers at the time. I have not read od any resentment from Kristina towards her half-sibling, who made a good career in the army and as an administrator, was ennobled and raised to the highest rank of the Swedish nobility and to riksråd (realm high councillor). They disagreed on policy though,

It seems that with time, Kristina moved ut from the shadow of her father, which she had not seen since before she became 4 years old and forged her own image - although she and her administration still held a good understanding of propaganda. Her first four years were marked by the last part of the 30 years' war, and she was consistenly molded in the image of Minerva, the Greek godess of hunting, war and victory, as the victorious queen of the north in the propaganda.

Sources:

"Den oövervinnelige" (the undefeatable) and "Ofredsår" (Years of war) by Peter Englund.

Historia kring trettioåriga kriget (History about the thirty years' war), in aprticular the section on Swedish propaganda by Sverker Arnoldsson.