Why am I so much slower indoor vs on roads??!!

Accurate distances indoors? Or the air quality may be bothering you.

Yes, core is bad for your running.

Because indoor tracks have a sharper curve that hurts speed.

If you really need some reassurance about your indoor track times, you can convert your times based on the size of your track.

http://www.ncaa.org/sites/default/files/NCAA+Indoor+Track+Facility+Indexing+Conversion+Summary.pdf

However, I wouldn't worry to much about it. I'm also not sure why you're comparing your road times, and indoor track times. Like why are you doing work on both surfaces that would be comparable?

The temp might explain it. Up to a certain point I'm faster the colder it is. My body just doesn't handle heat as well. Another explanation could be mental: Indoor tracks are boring places to run, and it's easier to run well if you're enjoying yourself. Other explanations: dry air, sharp turns, you might not be going as fast as you think you are on the roads.

I would say that if you enjoy running outside more, then do that. I have to race indoors, but do as much training as possible outdoors. There's a Lydiard quote about how enjoyment is key to the success of athletes.

Core sure won't hurt, and it could spell improved form and less injury risk as a consequence of the more stable torso core strength will give you.

Probably inaccurate distances outdoors. You aren't running as far or fast as you think

Acceleration is a vector( straight line), so you will go faster going straight than going around a curve. With indoors, you have to endure double the amount of curves to run 400 meters, that will curb your speed, causing you to run slower.

Also, you will exhaust more energy running around a curb than running in a straight line, because you have to run twice as hard around the curb to run at the same speed while running in down a straight away