As far as I know (but you probably want to hear it from Epic employee) you can use Unreal Engine free of charge in any project. Only if you have an income from UE4 projects you have to pay 5% royalties.
From FAQ :
How much do I have to pay for Unreal Engine 4?
UE4 is free to use, with a 5% royalty on gross product revenue after the first $3,000 per game per calendar quarter from commercial products. Read the EULA FAQ for more details. Contact us if you require custom terms.
About the logo, you can read about it on this page.