Ok, well, there's something you can easily do to narrow down your problem, and that's to clean your Mass Airflow Sensor (MAF). All you need is a can of electrical parts cleaner or non-clorinated brake parts cleaner, available at any auto parts store for a few bucks.
Here's a great illustrated writeup from jesda at Q45.org:
http://www.q45.org/maf.html
If that solves your stalling problem, you're back on the road. Jesda says to use an entire can of electrical parts cleaner, but I've done mine with just a couple few-second sprays of brake cleaner. The MAF begins to fail because its "burnoff" cycle starts to work erratically due to corroded contacts in the connector, so be sure to get at the contacts buried in the connector with something skinny and sharp, to scrape any off corrosion - again, something easy to do. Put 'er all back together after letting the MAF THOROUGHLY dry out and take it for a spin.
But if your pump is buzzing, you're gonna have to replace that soon. If you don't, it'll fry the Fuel Pump Controller Module (FPCM), which is another few hundred bucks. The pump and labor shouldn't cost you much more than $400 at a decent shop. I had mine done at a Nissan dealer for that amount. If the place you take it to has not worked on Q45s before, you might print out tangalora's wonderful thread for them, and be sure to warn the tech about inadvertantly bending the float during installation.
Something else that has been discovered to cause your symptoms is temperature surges from the transmission, which confuses the ECU/TCU. I mention this because you very recently dealt with your tranny. One poster dropped his tranny pan and cleaned the filter and his stalling problem mysteriously disappeared.
But try that MAF cleaning first - cheap, fast, easy.