Ding! I suspected a new girlfriend before getting to where you mentioned it. If he has another child now (assuming it is his and the GF's joint child), remember that child is your boys' brother or sister. That might change the situation a bit, in that he now sees what he's been missing out on and wants to reconnect and have a closer family that doesn't exclude his two eldest children. I'd put money on it being the girlfriend, but maybe??? he really is a "changed man" and things will not be so tumultuous for your boys and visiting their father, going forward; surely it is rough on children to not be able to know when they're going to see their parent, or to have that parent be unreliable. It is important to separate out the impact of his flakiness on you to the extent that it doesn't affect your children--sure it's embarrassing to have daycare take him off the emergency contact, but if the boys don't know and aren't negatively impacted by it, you might consider it water under the bridge.
As long as you show the Court that you're not interfering with your childrens' relationship with their father, you should have a fairly easy time securing court orders for reasonable things you think to be best for your kids. A step-up plan is exactly that--it shows that you intend to accommodate generous time between the children and their father. If he cares about your DSs at all, he will heed your advice about getting to know them before taking them alone. It is solidly in their best interest to know their father but not to just disappear to some place they don't recognize without any familiar adults. This is not to say it will be stress-free (to the contrary) but I wouldn't be overly concerned that you'll be ordered to send the kids for overnights with someone they sincerely don't know. I wouldn't normally say it was a good thing to not allow a parent to take a child (absent orders to do so) but in this case, letting him take the boys on his own sets the bar--if it's OK to take them today, your argument that they shouldn't be able to go with their father alone without a step-up plan, wouldn't hold water. Be a broken record in saying "yes, where would you like us to meet you and when?" to demonstrate that you are not interfering, but trying to help your children become acquainted with their father and strengthen their relationship.
Don't hold out hope that he will be prevented from being granted generous parenting time just because he owes a significant amount of back support (though some jurisdictions may disallow a parent who owes back support from petitioning for additional parenting time), though it definitely doesn't show well of his ability and desire to parent if he can't bother with that aspect of being a parent. A Court probably can't (again, depending on the laws in your jurisdiction) tie the two issues of parenting time and child support together, but that doesn't mean knowledge of his flakiness on all aspects won't be in their mind a little bit and potentially tip the scales to make them err in favor of your reasonable requests.
As has been mentioned, include specific times in your proposed plan. If you feel like specific hours each week are too rigid, you could also set a deadline (for example, the prior Thursday at 6PM) when he must notify you of his requested time else it defaults to X o'clock Saturday. Especially if you will meet away from home, but even if he is to pick up DSs at your house, you might consider being specific about what time (or how many minutes late) you can consider him to have forfeited the visit. That is, you shouldn't have to sit at the park for 4 hours waiting and waiting for him to show up to avoid him saying you blocked the visit; once he's 20 minutes late, you can go home. Regardless of the outcome (whether there is a visit or not) you should keep a journal of your interactions with him regarding the visits, the visits themselves, and anything noteworthy your boys have to say afterward. A spreadsheet ledger with date, what time he arrived, when the visit concluded, and additional notes (like agreed to shorten visit so Father could go to work, or, Father did not attend visit), is sufficient. Don't editorialize but keep the journal purely factual. This will be your evidence to the court if the issue needs to be revisited (and for the review he's requested already, so start documenting).
The trick your attorney should help you with is to make the step-up enforceable (to protect you and your DSs). On one hand, specifying exact dates for the phases, if he is doing--or not doing--something indicated in the Order, it is clear whether he is adhering to the Order or not. The drawback is that he might skip all of phase 1 and show up 2 months from now expecting to be entitled to phase 2 because the Order says that's when he gets to take the kids by himself. Ideally (that is, without courts) you would agree that he has to make X number of supervised visits or that the boys reach some milestone (like they say they're ready) before he moves on to Y number of mini visits before he moves on to Z number of longer visits, and so forth. The problem with this arrangement is that if he misses all but one of X supervised visits, it's your word against his and costly fights in court arguing what he did and didn't do or whether or not his partial visits count, to enforce whether he's entitled to the Y mini visits yet. Your attorney should have ideas on how to address this. I hope you have already or will retain a quality lawyer especially if he has retained a lawyer. These issues are too complex to navigate alone and critical mistakes can be made that will negatively impact your children's future. If you haven't retained a lawyer, you should meet with several (they usually offer a free hour consultation) to find one with whom you feel comfortable and confident. Even if you're sitting back waiting for him to follow through on action and choose to wait on paying the retainer, it is good to know exactly who you will call in the event you need to take fast action.
IFF there is any kind of statutory reduction of child support for the amount of increased parenting time/visitation he will (eventually?) take on, I don't think it should apply until he hits the end of the step-up phases. So if the goal is for him to have 20% of overnights by June 2014 that would entitle him to a 5% discount of parenting time (random, made-up numbers because the deduction amount is usually more complicated) that he doesn't get that 5% deduction at the beginning when he's only having short visits but no overnights, it applies in June 2014 when he starts having that many overnights. Perhaps you could even get no reduction ordered until he actually reaches that phase (that he would have to petition the court for the change) though I'm not optimistic that would be allowed--it will vary depending on your jurisdiction's laws. I doubt if back support would be affected, regardless of parenting time he assumes going forward. In any event, there should be an opportunity for him to not get the reduction he seeks unless he follows through on the visitation time also.
If he has another child that he is supporting (if the child lives with him) there may be a statutory reduction of child support on account of the support he provides to that other child. If that is the case, and the other circumstances that would contribute to the amount of child support he owes each month have not changed since child support was last calculated, it is conceivable that the amount he will owe each month could go down. If his income has increased over the past 4 (?) years, if your income has decreased and your state considers your income in its support model, if your state has adjusted the statutory amount due (this tends upward for each income level), if he's not providing health care insurance and was receiving a reduction of CS in order to provide health insurance, if your children's need for support has increased (you mentioned surgery), or numerous other factors could lead to the amount of his child support obligation increasing. It is stressful, but he may have opened a can of worms he doesn't like.
He can still withdraw his request to the Court, so if you can demonstrate to him that these shenanigans won't benefit him (i.e. if his true motivation is only to pay less child support and he sees that he will owe more) perhaps you can squash the issue with knowledge too. Look up your state's child support guidelines to get an estimate if the current statutory amount is what he's already paying or if it will likely change and share the result with him if it is unfavorable for him. Many states have calculators online to use for estimate purposes.