Batch Split Question

I am planning on splitting my next batch into two separate fermenters. After the batch has been in Primary for about two weeks, I plan on taking approximately 2.5 gallons and putting it into a glass carboy to soak in Oak chips.

However, I was planning on leaving the other 2.5 gallons in the primary for another two to three weeks. Will the excess trub from the original 5 gallons cause any issues with the remaining 2.5 gallons (Seeing that there is more trub for the remaining volume)?

I don't see that having much effect. Trub is trub, cold crash it if you want to make sure the undesirable stuff stays on the bottom when you bottle.

Not a problem.

Is cold crashing typically performed right before bottling/kegging?