I had a similar problem with my 7880 a few months ago. I believe the two problems you’ve listed are related to a failure of the PUMP CAP ASSY. (See page 298 of the Epson Stylus Pro 7880 Service Manual). Here’s a link to the part from a Chinese distributor: http://tinyurl.com/ybt6sg7u
When I replaced mine, I also replaced all the ink tank valve assemblies. I suspected more than one was not air tight. With these two simple repairs made, all air gaps in the ink lines cleared up and I got perfect nozzle checks. At that time, I also took the opportunity to lubricate the nylon drive gears using the proper lubricants according to the procedures in the manual. I was amazed at how much more quietly and smoothly the printer now runs.