These answers are very easily found by checking the tweaks thread, and using the forum search engine, respectively.
The issue with the file manager (an application called Thunar), is that deleting stuff simply moves the files to the trash, and the trash is on the NAND. Try right-clicking the offending items, holding down the Shift key (or the left shoulder button, which does the same thing), and then clicking "Delete" in the menu that popped up when you right-clicked, whilst still holding Shift. When a dialogue-box pops up to ask if you really want to delete the file (it does this to warn you that the deletion is permanent, and bypasses the trash), let go of Shift, and simply click "Ok".
I know how to delete the problem is when you trying to copy and delete large files about 100mb+ it comes with error and then it won't delete.
As for your zip file issue, try double-clicking on the file. You'll get asked what program you want to use to open this type of file in the future - look for "Squeeze", and choose to use this. That's the built-in program for dealing with zip files, and some other compressed filetypes.