Ok, that' good that you have the charging order in place. I had anticipated your hearing would be sooner than September as I presume that hearing is just a preliminary hearing to decide if there are grounds to appeal or not. COVID is causing delay in a number of courts right now.
Yes he would have to request a stay but in any event the court decides whether to order it. The order is only stayed if so stated within the order following his submission of an appeal notice.
Yes a re-hearing will only be granted if his appeal is upheld. This is a rare scenario there would have to be a serious procedural irregularity in the Final Hearing for this to occur.
If you get to court in September with no skeletons from either side then I suspect the court will throw the appeal out. The court may send an order out before then dismissing the appeal. The judge shouldn't give him more time unless there is a very good reason why he hasn't complied. If he can't comply he is expected to make an application to the court before September asking for an extension.
If he files late, you can file late if you need to. It's courteous to try and agree extensions with the other side first and to let the court know by letter/ e-mail.
In fact it is the responsibility of both parties to work together and try to ensure that a matter is court-ready. The duty in this regard is less if you are unrepresented. The Court won't be impressed with your actions if you just sit on your hands until September.
The reason I suggested speaking with the court was also because if he has filed a skeleton but not sent you a copy then they can send you a copy.
The appeal/ application for permission to appeal will be going to a more senior judge. They will likely be less forgiving of mistakes so if he hasn't filed anything by September it's very unlikely it would delay matters. The judge will most likely dismiss the appeal.