You asked about three questions that would each require at least a ten minute training video each.
Video size is the easiest, just scale it till it fits your sequence size.
If the video is really badly exposed it may not be fixable.
But.. I dont reccomend this but there are some auto filters in Pr that are simalr to doing "Auto levels" in PS. For your case perhaps the auto contrast. But its usually better to correct it manually with the color correction or levels filter than using the auto filters..
I know how to manually use the color correction in Photoshop, but how do I do it in Pr? And how do I apply it to the whole video?
Regarding the scaling.. How do I do that? Should I do it in Photoshop?
For color correcting a backlit scene like this, try Premiere's Luma Curve filter. It is similar to the Curves filter in Photoshop, but only affect the luminance of the clip. This may do better than RGB curves and Levels because it will not alter the color saturation of the clip as greatly, which can sometimes be problematic when doing large contrast adjustments. Additionally, the Luma Curve will provide much more control than the simple black/gamma/white controls of Levels.
If you need to do color adjustments (such as removing a cast caused by lighting confitions or incorrect setting of the camera's white balance) I suggest applying the Fast Color Corrector filter to the clip also.
I assume you were talking about audio noise reduction (and I'll get to that in a minute) but your video may require some VIDEO noise reduction -- especially since you are planning to significantly brighten then scene, which can make noise more apparent. Unfortunately, Premiere does not really have any suitable tools for this. There are some third-party options that other may suggest for this purpose.
As for background AUDIO noise, there are filters for this in Soundbooth and Audition -- but it is not likely you'll have much success in removing this noise unless it is constant (such as 60Hz hum, tape hiss, line noise, a fan, etc.). 'Random' and intermittent noise (such as noise from public places) is nearly impossible to remove, though you may be able to apply some EQ to make it less distracting.
Thank you that was very helpful. The fast color correction actually worked better for me, because playing with the Luma curve just makes everything too bright. I managed to do it with the fast color correction by choosing the black points.
Regarding the audio fix, how do I apply the EQ? Are those presets that available on Soundbooth/Audition? Which one specifically should I use?