Hello. I believe I can be of help to you with this issue.
It sounds as though you are doing many things quite correctly to build a relationship with this boy and compensate for the loss of presence of his father. I'm wondering if you could comment on the nature and quality of relationship he has with his mother. Specifically, how much one-on-one time is she spending with him each day e.g., play sessions, reading, exercising. Day care of course is no substitute for having an adult attachment WITH the child during the course of the day so many kids don't do very well in day care, especially if a divorce has occurred.
Every hypothesis I come up with based on your post suggests that this boy would be helped by one-on-one time and attention in activities that would provide social reinforcement and rewards, but would introduce elements of gradual frustration that he would have to learn to deal with, in small doses, then somewhat larger doses, etc. So building things that can fall over and 'break down', working hard on tasks that sometimes succeed and sometimes do not, will help. When is not feeling stressed, purposefully setting up situations that require him to delay gratification is important. His mother should do this with him at least a few times each day. When he asks you to play a game while you're you or his mother are the middle of a task, have him wait for a short period of time. Praise him for being patient, and gradually work on increasing the time he has to wait. For instance, if he is hungry you can decide to bake a half-dozen cookies from pre-made cookie dough you buy in grocery stores. He can help put out the cookie dough onto cookie sheets and then back them. When they come out of the oven, he will have to delay the urge to eat one because they are hot, but this is an opportunity to use the cookies further to help delay gratification. For example, "We need to clean up the kitchen counter before we eat a cookie" or, "lets put the cookie on a plate here and watch it for minutes before eating it"; let's stack up the cookies and so they can wait a few minutes before we eat one.
If he is dealing with a frustrating situation, alway be sure to empathize verbally, "You worked so hard on that building and it keeps falling over," validates his feelings. Verbalizing the problem can also help pull for possible solutions. After empathizing about the building, saying something like "How can you make it stronger so that it will stay up?" gives your child a cue about solving the problem.
Start to brainstorm other opportunities to require gratification delay and consideration of alternatives. For example, you can purposefully set up situations in which you want to say 'no' to this boy about something; for instance, say 'no' to one request, say, for a toy or object, and then, provide an alternative for your child when he can't have a desired object. When your child asks for candy in the grocery store, suggest another, more acceptable alternative. For other children, however, the alternative choice may be reasonable. If your child is engaged in an activity that is causing frustration, suggest that he walk away for a few minutes or try to engage him in a different activity for a period of time.
All in all, this boys sounds somewhat developmentally or 'learning' delayed in both his emotion regulation skills and some socialization skills. This typically means that parents haven't really spent enough time with the child doing the sorts of things I mention above. Some kids need more of the kinds of training activities I mention above than others. The above activities can be built into play sessions with friends i.e., invite another child over and play with the two of them and monitor, coach and direct their sharing behavior; verbally praise and reward episodes of reciprocating and sharing. This is the part of the one-on-one stuff I mentioned at the beginning of this post.
I'm going to pause here and solicit your reaction. My point is that it doesn't matter what this kid's history is, whether he has biological deficits or problems, or learned misbehavior. The only way to remedy any of these is through more intensive parent or adult child training and exposure to repetitive situations that can help build emotion regulation, frustration tolerance, and delay of gratification, as well as sharing skills. Now his biological father needs to be doing some of this stuff with him as well; most men though, don't have a clue about exactly what to do to be helpful to their child so some coaching is necessary e.g., from a child behavior therapist or someone known as an applied behavior analyst----they understand the sorts of things I'm mentioning above and can coach parents. What do you think?