Like Lyonaria said, I would file at the end of July just to be safe and apply priority. UKVI are very particular.
Also make sure that he has a letter from his employer (also dated within 28 days of your application) that explains: what type of employment he has (FT/PT/Permanent/etc), his annual salary, the date of the start of his employment and some other things.
Regarding the bank statements, the July bank statement will have to have a deposit of his July payslip on it.
For example, my SO gets paid on the 28th of every month. I had to wait until the 28th to make sure that his latest payslip was on the latest bank statement.
Your husband might have to mail them to you, which is par for the course with what I had to go through. I had a huge envelope of paperwork mailed to me from my SO.
Also, make sure that they are either the original bank statements; or if you have copies, that they are stamped (on every page) by the bank or accompanied by a letter from the bank manager. You might need to order the original statements from your bank so that's something you'd need to tell the bank in advance.
On the refusal, you do have to report that on your application, and I would definitely put that paperwork in your application this time. Also for the future applications, make a copy of those refusals for your own personal record. You will need them for other visa applications.
I have no idea about the unpaid hospital bill as I have never been hospitalised (knock on wood) so I don’t know how to handle it.
Again, I would make a separate thread for this stuff because the fantastic people on this forum will give you super detailed answers. The people on here are wonderful and will try to help as much as possible.