It’s entirely possible now, but in the past, no. Well actually it is possible now, considering Oryx is dead.

Honestly i doubt it. Oryx was the literal king of the hive and taken. He outsmarted and killed savathun and xivu over and over again...to the point where they had to leave his side to become stronger. Now that Oryx is dead, sure why not. Bungie does need a plot after all.

Absolutely. She clearly has the power to “Take” and has used it on a much more grandiose scale than Oryx ever did. Going as far as taking her sister’s children (Oryx did take Malok but that’s all we’ve seen when it comes to Taking other hive gods) She tricked the worm God’s with a black hole or something of the sorts. [spoiler][/spoiler] In the last wish raid, it is shown that Riven, an ahamkara was taken. Savathun has quite clearly, conquered the Dreaming City, the fight for it is essentially pointless. Not to mention, she has certainly found more tricks for using taken blights. (Via watching us through them. Etc) And is certainly way more aggressive and organized with the Taken. After all, Oryx wasn’t really able to pull any major offenses, aside from Mars, which took up the last of his resources (temporarily), and the Reef’s failed attempt to fight him. Savathun has certainly had lots of time to grow her army, if I were Zavala, or Mara Sov, I’d withdraw any guardians or Corsairs from the Dreaming city and organize some form of quarantine for the taken. And also keep an eye out for Savathun. Sorry for the long post.

She already is. Changing the rules of an entire city on a regular weekly basis? Creating a time loop? Yeah.

If she isn't yet she soon will be. That's the whole point of the time loop in the Dreaming City. By fighting Dul Incaru and her minions every three weeks we're generating a potentially infinite amount of power for Dul Incaru to tithe to Savathun. If Guardians continue the cycle without finding a way to break it Savathun will rapidly become more powerful.