Booting a 32-bit OS in EFI mode on a system with a 64-bit EFI is difficult, but not impossible. Since it's a fresh installation, though, and since it's a hassle to do the cross-bit-depth booting, I recommend you re-install using a 64-bit Ubuntu.
To reduce the odds of making the same mistake, go into your firmware and disable the BIOS/CSM/legacy support. This support is normally disabled by default, but a lot of sites provide the misguided advice to enable it as a matter of course when installing Linux. The result is problems like yours. If your boot medium is prepared correctly, it should boot fine in EFI mode and you should be able to re-install with few or no problems. (Note that some tools omit EFI boot loaders when preparing USB flash drives. Rufus tends to do a decent job, provided you pick the option to prepare an EFI-bootable disk.)
As to your video problem, that's probably (but not certainly) unrelated to your EFI-boot problem. I recommend you try a 64-bit EFI-mode installation and, if you continue to have problems, search for questions on this topic. Video problems are not uncommon, and chances are your problem has been encountered before and overcome. Knowing the video chipset may be critical for finding a solution, so if you need to post another question, be sure to include that detail.
One final point: non-LTS releases are supported for just nine months. The 15.04 release you're trying, in particular, will reach end-of-life (EOL) in January of 2016. Thus, I recommend you try either the latest 15.10, which will reach EOL in July of 2016; or 14.04 LTS, which will reach EOL in 2019. If your video hardware is brand-new, 15.10 may be a better bet, since it's got a more up-to-date kernel and X server; but of course the extended support period for 14.04 is generally desirable.