Were there any charges legally/morally? How big was her role in the crimes commited? How was decided which persons faced the trial?
What exactly is a moral charge? I don't meant to sound rude, but the idea of a 'moral charge' in a legal trial is somewhat superfluous - either one is charged under a law or not.
Braun essentially had nothing to do with politics despite being part of the inner social circle around Hitler, and allowing her to have a role would have come up against basic Nazi ideology about the place of women in society and politics (pretty much none for the latter).
The process of defendant selection for the Nuremberg Trial was through a process of inter-Allied negotiation. The US was the driving force behind the trial and formulated the charges and concept of conspiracy in late 1944/early 1945. The specific selection process began really with the London Conference in June-August 1945 involving the US, UK, USSR and France. Just before the conference, US and UK officials had informal discussions in which a rough list of 16 names was informally agreed upon*. Selection went on the backburner however when the conference started and the difficult issues around negotiating the charges were played out. The second week of August saw selection come back under discussion, and by the third week of August a definitive list was prepared. This included all but one name from the June list (Hitler), and the French and Soviet delegates insisted on a further 9 names being added making 24 in total.** This final list was supposed to be published on August 24, however at the last minute the Soviets insisted on a further six names being added, of which 2 were accepted (Raeder and Fritzsche)
Long story short, it involved diplomatic and political discussions between four nations to decide who was to be tried.
*Hitler, Schacht, Seyss-Inquart, Doenitz, Funk, Speer, Goering, Hess, von Ribbentrop, Ley, Keitel, Streicher, Kaltenbrunner, Rosenberg, Frank and Frick. ** Above minus Hitler and plus von Schirach, Sauckel, Jodl, Bormann, von Papen, von Neurath, and a member of the Krupp family (unclear who was intended).
While phoenixbasileus did an excellent job explaining why Eva would not have been present at the main Nuremberg trials, it is worth noting that she would not have simply been able to go on living like nothing had happened.
While it is unlikely she would have been involved as a defendant, there were also Subsequent Nuremberg Trials for lesser Nazi criminals. Again, it is unlikely she would have been put in front of a judge, (Though this cant be 100% ruled out, considering how vague some of the charges are, like "Planning a war of aggression") but her proximity to Hitler's inner circle, and her important status means she would have almost certainly been detained and involved as at least a witness. If I had to guess, she would have fared similarly to Traudl Junge, Hitler's personal secretary. She would have been detained, questioned extensively, and depending on whether she was in Soviet or Anglo-American hands perhaps subjected to a certain amount of cruelty (or even rape in the case of the soviets) during this process.
Besides that, there was also a much broader Denazification by both the allies and the new German government. This covers a broad range of practices, from "in house" secondary trials by Germany (which convicted a number of people released at Nuremberg) to forced cultural changes. It is here where she would be in the most danger of facing jail time. Another likely event at this stage would be the confiscation of everything she acquired during the Third Reich.
That is about all I can offer. Ultimately this is all speculation, and we will never know for sure.
If there is any comparable case, it might be that of Hedwig Hoss who was the wife of Rudolph Hoss, the commandant at Auschwitz. Obviously, she profited enormously from the Holocaust and had a decent enough sized estate with gardeners, chauffeurs and the like taken from camp inmates. Obviously, she did not get to keep any of the trappings of her former lifestyle and was plunged into poverty, but she was not charged with any crime.
What would the Soviets had done to her if they had captured her?
Please do not ask me for a cite, but I recall Hitler's (half?) sister Paula survived the war and ended up in Russian hands. She was questioned for some period and then released.
Eva of course would be a source of a lot of interesting nazi gossip, but few would think she had any guilt or could serve as a rally-symbol for un reconstructed Nazis. I presume she would have been questioned, released and watched carefully for years.