First, you have to make sure that you do not have both constructs in the same plasmid. Digest the plasmid to know if they are in the same plasmid or in separate plasmids.

If you think there is a mixture of 2 different plasmids. One way to solve this issue is to digest the plasmid mixture with an enzyme that cuts the plasmids only once (a unique cutter).

Run digestion mixture on gel.

Gel purify the correct size plasmid.

Ligate the plasmid again (easy self ligation).

Transform the circularized plamsmid. Now you should have a the palsmid of the correct size that contains only one copy of EYFP.

This is not possible. You can't have two different plasmids with the same origin of replication. One will inevitably outcompete the other. Most likely it means your PCR reagent is contaminated with the original template. Try some other method, like restriction digest or sequencing.

I would suggest re-checking your primers for colony PCR; it is likely that one of them misprimes on an additional site. I use a combination of primers (one priming withing the insert, one within the vector backbone) that can anneal at 58 (or even above).

As mentioned by John Schloendorn you cannot have two plasmids with identical origins within the cell. You could, however, have two clones growing seemingly like a single colony. You can re-streak a clone that gave you that double band and do PCR on a couple of resulting colonies. I am, however, skeptical, that this could have been the case with all colonies you tested.