How much of a co-ruler was Theodora (500-548) to her husband Justinian (482-565)?

So, Theodora is one of those subjects where pop-history has a consensus that actual history doesn't. To be frank this pop-historical consensus is made by cherry picking good things from sources and discrediting anything negative. For example, the same source that mentions her as a ruler, perhaps even better than Justinian, also accuses her of murdering her own children so Justinian wouldn't set her aside. Setting aside what this says about her character, that action clearly implies Justinian had more power in the relationship. The justification for including the former and not the latter is that the latter is slander and the former is reporting a true fact because... well, it's not actually justified.

You brought up two common ideas: that she was a co-emperor and that she prevented Justinian from fleeing the Nika Riots. The idea she was a co-emperor has two pieces of evidence. First, from artistic representations where she was often portrayed more prominently than ministers, and sometimes equally prominent with Justinian. While this is unusual, it wasn't unprecedented and doesn't directly imply a specific title. What is known, though, is that it was often Justinian who was commissioning these pieces and that it actually became more common after her death.

The other piece of evidence is a mention by Procopius that she was crowned as his equal. This could mean she was raised to that title. It could also mean that Justinian didn't force her to make symbolic acts of submission and had her crowned rather than treating her as simply a consort. Considering the lack of coins and inscriptions using the title, it's likely he meant the latter. Even Procopius never describes her as acting as his equal or exerting direct authority the way a co-emperor would. There are some historians who do believe it's literal but the weight of evidence is that, while she was highly influential on her husband, Justinian was really the one in power.

Secondly, the Nika Riots. Again, the specific story and speech is mentioned in only one source: Procopius. And we have no reason to believe Procopius was actually at the meeting. The language Procopius uses in describing Theodora is very clearly meant to evoke Sejanus, right down to using a Greek version of Sejanus's famous title 'socius operis'. And like Sejanus, Theodora is described as setting her own policy, thwarting her partner, starting reigns of terror, and taking on imperial powers. The incident might have happened but it serves an invective and paralleling purpose in the work.

The fact that Justinian is considering fleeing is cowardice since he still has his armies. That Theodora needs to encourage him to even fight for his throne is a clear insult to Justinian. It implies Justinian and his ministers are weaker and more cowardly than his wife. That, along with the excessive butchery of suppressing the riot along with the executions of innocents at Theodora's insistence, makes her a callous butcher and is another accusation of murder. (Side note: As to her being co-emperor, you might also note her famous speech begins with her saying that she doesn't usually attend the council of ministers that runs the state...) This also parallels Drusus and Sejanus' purge of Pannonian mutineers, comparing the couple unfavorably.

So while the incident might be real, the account we have of it is perfectly constructed for a specific literary purpose. And Procopius is not above manufacturing incidents wholesale.

Sourced from all three works of Procopius (The Secret History, The Wars of Justinian, The Buildings of Justinian), The Cambridge Companion to the Age of Justinian, Women in the Byzantine Empire, and The Secret History and Propaganda.