Looking through your post I am wondering if you are getting some of her cues mixed up. It looks like she is eating small amounts really often. So basically it sounds like she is snacking. I know at 7wks my dd was eating at least 3-4 ounces per feeding, and able to comfortably go at least 3 hours between feedings. At 7 weeks dd was staying awake anywhere from 1.25-1.5hrs before needing to nap. Before starting easy (right at 7weeks like you) dd sounded a lot like yours - lots of snacking and really short naps - 20 min, and only napping well in her swing. Also, I have heard that eating frequent small meals can actually make gas and reflux worse, and that may be happening to your dd.
So I started with a pretty firm EASY - only fed her every 3 hours unless she just couldn't stand it, and was OBVIOUSLY starving. I also basically abandoned the swing. I only let dd swing when she was awake and in no danger of falling asleep in the swing. I made sure we were home for as many naps as possible, and had dd take them in her bassinet every time (and eventually transitioned into her crib around 3mos). I knew one day she would grow out of her swing, and it would NOT be pretty if that was the only way she knew how to go to sleep.
I also established a really good wind down routine so dd knew exactly what was happening next. Ours was: diaper change, swaddle, turn out lights/shut blinds and curtains, turn on sound machine and monitor, sit and rock and sing the same song, till dd was asleep, put her in bassinet. I very gradually went from rocking dd to sleep to laying her down more and more awake until I could lay her down wide awake after our wind down and she would go to sleep on her own with very very little fuss if any. Now, dd is an angel baby and so this worked for us - I never really did use pat/shh, but many people use it and get lots of results from it. I just never had to
So I hope some of this helps. I think the main thing I would do would be to ONLY feed her every 3 hours unless she just seems ravenous, and work on getting her to nap better. I hope that gives you some direction - post back and let us know how your getting along!