I know its been a few months since this was posted, but if you haven't already been to Phoenix to find out for yourself, I can share a bit.
I have worked for Phoenix, and never had a problem with them. They are fairly well established in Kyiv, have a fairly good body of students. They can be a bit demanding on their teachers (professionalism, teaching ability, how well can you sustain your classroom/how satisfied are your students with you), but at the same time will invest in you if you show willingness to learn (and learn quickly). If you are a good teacher, you will have no problem with them.
In regards to pay, while I was working for them they fell into a bit of a grey zone (you were paid as a specialist 'under the table,' a practice that almost every language school did/does-- not sure how it has changed in the last two years).
Expect mostly part-time hours, evenings (12 to 18 hours a week at $15US an hour-- enough to live on but not enough to save up). Good experience though.
ELC is another school to look into. They have been around a bit longer. Not too sure where to point you for full time teaching (Denis School maybe... everything else full time demands strong credentials and experience, and I am not sure where you are on that).
