I STRONGLY suggest looking into hiatal hernia - this prevents the valve between esophagus and stomach from properly closing and you can regurgitate some of the stomach contents as you describe. You want to have this fixed cause it can be eroding your esophagus lining. I believe there are chiropractors that can help this problem along with some dietary changes and supplements; just do some research here.
It is possible that some of the greasy bad tasting stuff coming up is bile.
Yeah those Antibiotics you take for your urinary infections are wiping out the good flora in your gut which allows candida, bad bacteria and parasites to thrive - you want to make sure you replenish the flora with plenty of probiotics or fermented foods.
Mike