You are confusing dissolved oxygen and oxidation. Your starter contains both. The beer will consume all of the dissolved oxygen in the starter. However all of the oxidized malt flavors will carry over to the finished beer.
The ideal is to rack warm into a co2 purged vessel, then crash cool under co2 pressure. If you are racking into a co2 purged vessel and keeping o2 away from the beer when you rack, it does not matter if the beer is warm or cold.
Unless you are cold crashing under co2 pressure, you are doing a LOT more oxidation than racking warm. Cold crashing sucks in oxygen, and cold beer will absorb a lot more gas than warm.
The keg is like a big bottle. Once the beer is carbonated properly, you can disconnect the gas and it will stay carbonated (as long as there are no leaks).
You are confusing dissolved oxygen and oxidation. With hot wort you will have less dissolved oxygen, but oxidation of the malt compounds happen faster.
Also, make sure to remove chlorine from any water used for rinsing and/or sanitizing. I have tasted beer made with RO water that got a bandaid taste just from using chlorinated water in the starsan.
Any time you cold crash in a carboy/bucket, it will pull in O2. As the liquid cools, it absorbs more gasses into solution. Unless you are cold crashing under co2 pressure, you are oxidizing your beer. Consider dry hopping and cold crashing in a keg under co2 pressure.
I always used the beer gun for bottling, until I won one of these in a competition:
https://www.amazon.com/dp/B00N3D9POI/?tag=skimlinks_replacement-20
I do not know that I would have spent almost $400 to buy this bottler, but now that I have used it I don't know that I can ever go back. Very...