I reside in CA, working remotely. The corp is located in NJ. I travel to NJ 31 days/year, but lodge in NY. 1) what states shall I file? 2) How do I split my income?

Since you're a resident of CA, ALL your income is taxable by CA, regardless of where you earn it.

NJ can tax you as a non-resident on the portion of your income that you earned by physically working in New Jersey.  Therefore you must file in both those states.

CA will credit you for the taxes paid to NJ, so you won't be double-taxed.  In TT, complete the non-resident state NJ return before you do your home state return, so that the program can calculate and apply the credit.

If you did no work in New York, then you have no filing obligation there.

In the Personal Info section of TT, enter your State of Residence as CA; answer No to having lived in another state; indicate that you had Other State Income from NJ.  This will prompt he program to produce the correct forms.  

My W2 shows NJ and NY tax withholding , because the NJ corporation register my address as NY.

How do I get the NY withholding back? 

To receive a refund of the NY withholding, you'll have to file a non-resident NY tax return on which you show the withholding but allocate zero income to NY.