Why φύγει and not φύγω
For the same reason that you might say "I have been" and not "I have am".
The helping verb έχω is conjugated to match the subject, but the next verb is in a non-finite form (with endings that don't change depending on the subject).
In English, that's the past participle; in Greek, it's the απαρέμφατο -- which looks like the third person aorist subjunctive (i.e. the form you would use in the future for "he will ...") -- in this case, φύγει just like in θα φύγει "he will leave".
(I've heard that the origin of this form may be the ancient Greek infinitive.)