02 Escape No Power Misfires Galore

My daughter's friend from college just limped into town in an 02 Escape desperately needing help. My back is up against a wall since it is exam week and I don't have long to get this thing running. Here's the deal: the vehicle has no power and is misfiring on 4 of the 6 cylinders. However, when I let it sit for a while and started it back up and drove it, it ran fine for about 15 minutes, at which time I kicked it into overdrive on a long straightaway...bad move. The car almost shut down and wouldn't go up even the smallest hill after that. When I put it in neutral and floored the accelerator, it would only get to 2200 rpm. Now it's running rough. The girl says it's been getting worse over a few months. At the same time, she says she is having to jump the battery a lot. Any relationship between these seemingly unrelated problems?

Normally, I'd just replace the coils on the individual cylinders, but I have 4 cylinders misfiring and these things are $65 each. Before I run down that path, does that sound like the issue? Are there other things that cause misfires -- vacuum leaks, camshaft sensor, crank position sensor, etc.?

I'd first test the battery and alternator. No electricity, no spark.

Update: I changed the spark plugs even though they looked great. I checked the battery connections and they were good. I changed the fuel filter...no improvement. I then unbolted the catalytic converter and voila! it's fixed. The converter was a shambles. $150 later and I'm good to go. Thanks for the suggestions.

Was this problem a plugged catalytic coverter, or a blown cat?
It sounds like a plugged catalytic converter choked your engine out, and caused misfires.

I didn't do an autopsy.

A blown catalytic converter would of thrown a P0420 or P0430 code ''Low catalyst efficiency'' , the engine should be good, but with poor emissions.

I actually had a plugged CAT causing the same symptoms as your daughter's, I took the cat guts out and added an aftermarket catalytic converter.

The OEM was $600, the aftermarket was alot cheaper.

Added a ACME AUTO EMISSIONS FILTER, to help protect the MAF, IACT, H02 sensors, Throttle body, and catalytic converters from Emission gunk.
Poor emissions can lead to poor fuel economy, performance, emissions, and bad catalytic converters.

see http://www.ideabuyer.com/ipavail/acme-auto-emissions-filter