"Φαΐ" and "φαγητό" technically have no difference. They both mean food, and they are synonymous with "τροφή" (which is the exact translation to food). Personally, I think "φαΐ" is a bit more casual.
"Τροφή" is often used with an adjective that describes consistency and doesn' have the element of actual cooking to it:
"Liquid/dry/canned food" - "Υγρή/ξηρά/κονσερβοποιημένη τροφή"
"Γεύμα" is the exact translation to meal, but it sounds a bit formal in Greek and it's not that commonly used. "Φαγητό" could refer to a meal as well, and it's used more often. Greek people use "φαγητό" when they refer to eating any kind of meal, like lunch or dinner (pretty often outdoors, at a restaurant or something similar.)
For example, "Θα βγούμε έξω για φαγητό/δείπνο" = "We will go out for dinner"
"Το φαγητό/μεσημεριανό είναι έτοιμο" - "Lunch is ready"
Also, "Μου αρέσει το πρόχειρο φαγητό - I like junk food."
I hope that's clear enough ^.^
No, unfortunately :P
The article can't be omitted in this case. It's always used, even in general statements of liking.
Μου αρέσουν οι γάτες - I like cats.
Μου αρέσει αυτή η γάτα - I like this cat.
This one isn't a word-for-word translation, I'm afraid. The meaning is general, but the article is needed in Greek.