You do realize that a cop doesn't need a warrant when ACTIVELY PURSUING a suspect, right? And they damn well can go right onto private property while, again, actively pursuing a suspect.
And her belief that the syringe may have been a weapon is absolutely fair in a place like Central City where all this shit has been going down, and she's currently facing a man who is supposed to be dead. ANYTHING could be a weapon in that situation. This is a city full of meta humans, for crying out loud. What's she supposed to do, assume the big syringe thing he's holding is perfectly safe and somehow psychically intuit that he's not dangerous or a metahuman?