Would the legalization of same-sex marriage raise income tax rates or increase the amount of government entitlement spending?
No. It hasn't done either of these on the state level in states where it has been legalized. Although many same-sex married couples will pay less federal income tax when the federal government begins to recognize same-sex marriage, the reduction in tax revenue is not large enough to create a need to increase the tax rate for everyone to compensate for it.
As for entitlement programs, marriage in general does not generally increase government spending in that area. Many entitlement programs have eligibility criteria that will actually eliminate more gay people when their marriages are recognized on the federal level and their spouses' income will be counted when determining eligibility. Benefits for married couples are not greater, and in fact are sometimes less, then the benefit for the same two people when they are considered as single individuals.
How would the legalization of same-sex marriage help the judicial branch?
The legalization of same-sex marriage would not provide any benefit to the judicial branch of government, unless you are referring to a theory that once the issue is settled there might be fewer lawsuits challenging the status quo. The difference in case load would be so minute that it is not even worth trying to figure out whether the judiciary benefit from having more work or less work to do.