Parliamentary immunity for MEPs does not extend to their privates lives. MEPs are only immune from prosecution for public statements they make (like, for example, "Qatar has a great human rights track record. It's the best, the greatest"), not for anything else they do.

I am not sure she had parliamentary immunity.

It seems like that only applies during sessions of the EP, which isn't in session right now.

She would probably be able to claim immunity to attend the session next week, but as the first thing will probably be to lift her immunity I doubt she'll bother

Having 600K isn't in itself an "act". And they must have some reason for mentioning a "Gulf state". I wonder if they've had her and the others under surveillance, and nabbed her while brown envelopes were passing back and forth.

She had the money hidden in her house. Her father was caught with a suitcase full of money.