Welcome!
Sometimes it takes a few days for the posts of first-time posters to show up. Our apologies for not noticing this earlier.
To your specific questions:
1. Given that the school has a program for identified gifted, it would be reasonable to put him on their radar, with very specific, skill-based descriptions of his current performance level and behavioral traits that you suspect to be gifted-related. (E.g., he adds and subtracts fluently to 100+, he figured out multiplication by repeated addition on his own, he can read Amelia Bedelia or Frog and Toad independently. He is better able to use his self-management skills when he is intellectually or creatively engaged by appropriately challenging tasks, such as...)
2. This is very much family- and child-dependent, but my personal opinion is that spending a half-day in work far below his true instructional level is better than spending a full day with such work. And you'll still have the half-day to spend one-on-one with your younger child. Other within-child factors can moderate this, though, as some children are so socially motivated that they are willing to tolerate the lack of intellectual stimulation for the sake of the social stimulation.
3. You didn't ask this question, but given the success you've had with homeschooling him for the preschool years, is that something you've considered (or are in a position to consider) continuing into kindergarten?
