διαγωνισμός always has the meaning of contest/competition. But ανταγωνισμός is when you compete against others and it has a kind of negative meaning. No you can not use it in this phrase. You could say however this : Υπήρχε θεμιτός ανταγωνισμός. Which has a very different meaning but I used it just to show you how that word works. It means that the competition was fair (you were competing against others but in a healthy/friendly way)
Not quite. Ανταγωνισμός is a direct translation to competition, and the difference between ανταγωνισμός and διαγωνισμός is pretty much the same as the one between competition and contest in English (I do know that these two are kind of interchangeable in english, -since you can almost always use competition instead of contest, but not the other way around- but they are definitely not interchangeable in greek, for kind of the same reasons).. Meaning that:
Ο ανταγωνισμός is more than one event (while ο διαγωνισμός usually is) and it is almost exclusively used to describe people's competitive behavior.
Ο ανταγωνισμός doesn't always come with a prize (while ο διαγωνισμός usually does).
I hope that was helpful ^.^