Are there other things we need to take into account when comparing July to September?
The summer tourist season is from late May to early September. Late September is approaching winter but a "good" September day may well be better than a "bad" July day even though the weather does get colder and stormier over the July – September period.
Any of you have experience with going hiking inland in September?
Sadly, not personally but a friend from school spent several weeks camping on his own in an isolated fjord there in the summer holidays.
How was it?
For a week all the conversation he had was with a donkey so it was boring at first but he survived and had a good time.
Was there snow already?
No, except on high ground.
For more than you want to know about Iceland's weather see Einarsson. May be worth bearing in mind Icelandic Met Office:
Iceland enjoys a much milder climate than its name and location adjacent to the Arctic circle would imply. A branch of the Gulf Stream flows along the southern and the western coast greatly moderating the climate. However, this brings mild Atlantic air in contact with colder Arctic air resulting in a climate that is marked by frequent changes in weather and storminess.
Also a special current warning:
Glacial water is flowing into Múlakvísl river, south of Mýrdalsjökull. High but constant conductivity is measured in the river and gas measurements in the area show high concentrations of hydrogen sulfide. People are advised not to spend extended periods close to the river bank.
Is it possible to stay in the cabins in September?
Yes, see for example Bungalo.