Like is the plot and scenes determined beforehand, or is it still possible to change details in the story while an anime is airing. If so are there any series?
I am assuming you are asking about the type of anime that is made by an animation company where the plot and story details are all made by the writers and directing staff.
The straight answer is Yes. The plot can be altered as the series is airing if it was not completed prior to its airing on television or it was affected by external factors.
A famous example would be Pokemon: Best Wishes. Where the plot was altered when the episodes Team Plasma VS Team Rocket had to be put on hold due to the Tohoku 2011 Earthquake. Originally the two episodes would have been crucial to the plot of Pokemon BW, the earthquake forced the production staff to change the plot of the series in order to avoid airing the episodes that depicted earth-shaking scenes. This is a case where the plot was altered due to external resons.
This episode was originally scheduled to air in Japan on March 17, 2011, but was postponed, along with the original BW024, indefinitely because of the Tōhoku earthquake and tsunami and the Fukushima Daiichi nuclear disaster. As of Team Plasma's Pokémon Power Plot!, its events have been removed from continuity. Unlike the only other skipped episodes in Japan, this two-parter was supposed to be important to the overall storyline instead of being a stand-alone episode.
However, these cases are rare and some of the time, the anime is being manufactured as it is airing, so the plot would likely to be undetermined until just a month or later before airing for really late production crew.
A good example would be Neon Genesis Evangelion, where its plot was changed from having 28 angels to just 18, counting the humans. This was mostly due to Anno Hideaki failing to complete scenarios on time, changing plots due to his new experiences as the show was airing which resulted in the production hell of NGE, and its plot change.
By episode 13 the series began to deviate significantly from the original story, and the initial script was abandoned. The number of Angels was reduced to 17 instead of the original 28, and the writers changed the story's ending, which had originally described the failure of the Human Instrumentality Project after an Angel attack from the moon. Starting with episode 16, the show changed drastically, discarding the grand narrative concerning salvation for a narrative focusing on the individual characters.