To start with your MacBook Air will not run Mavericks, so you can put that one aside as incompatible.
As for Yosemite, the earliest possible version that will run on your model MacBook Air is actually Mac OS X 10.10.2 (build 14C2043). My guess is that the copy of Yosemite you've downloaded is either 10.10.0, 10.10.1 or a build version of 10.10.2 earlier than 14C2043.
Regardless, your best option is usually to boot into Recovery Mode.
You can boot into Recovery Mode and reinstall macOS from there. You will have up to three options:
- Hold down the commandR keys as soon as you power up to reinstall the latest macOS that was installed on your Mac, without upgrading to a later version
- Hold down the optioncommandR keys as soon as you power up to upgrade to the latest macOS that is compatible with your Mac
- Hold down the shiftoptioncommandR keys as soon as you power up to reinstall the macOS that came with your Mac, or the version closest to it that is still available1
1 That third option is a new option only available since 27 March 2017 and requires macOS Sierra 10.12.4.
For more info refer to: About macOS Recovery
IMPORTANT - Your question doesn't specify where you've downloaded the installers from. Please note that if you haven't downloaded them directly from Apple then there is no way for anyone here to confirm the suitability/safety of what you've downloaded.
Since you're having problems with Recovery Mode I suggest you reset your Non-Volatile Random-Access Memory (NVRAM).
Resetting the NVRAM
To do this on your MacBook Air, follow the steps below.
NOTE: Before following the steps below, ensure you have no external hardware connected and make sure you use the built-in keyboard.
- Fully shut down your machine.
- Press the power button and then press the commandoptionPR keys. You have to make sure you press these keys before the gray screen appears or it won’t work.
- Hold those keys down until your Mac reboots again and you hear the startup chime for the second time.
- Let go of the keys
Once you've reset the NVRAM, power off and try the various Recovery Mode keyboard shortcuts again.