Shouting down other people deprives them of their freedom of speech, and is rightfully prevented. Padilla was detained because he was attempting to do that: disrupt someone else from exercising speech. He could have made the exact same speech in his own space without consequences.
If you disapprove of how Padilla was treated, that's fine, just be honest about why he was detained: not for the content of his speech, but his attempt to prevent another from speaking.
Interrupting or questioning people isn't a denial of first amendment rights. You're using extremely sloppy logic, mixing "freedom from interruption" with "freedom of speech".
Absolutely, repeatedly interrupting people with questions can get you arrested. Go to a public commentary session at your local town hall. Exceed your allotted time period and keep questioning the officials. You'll eventually be arrested and taken away. Because in doing so, you're depriving the rest of the town from their opportunity to give a public comment.
There's nothing complicated about this. Padilla isn't being treated any differently from anyone else. Freedom of speech does not entail freedom to prevent others from speaking.
If you disapprove of how Padilla was treated, that's fine, just be honest about why he was detained: not for the content of his speech, but his attempt to prevent another from speaking.