Thank you for your questions.
You mentioned that the six month period from your last trip to Europe ends on September 14. If you go over before 14 September, you would be entitled to use up the days still left of your three months entitlement.
You would then need to leave the Schengen area by 14 September at the latest.
To leave the Schengen area, you are not obliged to come back to the US. You can simply go to a non-Schengen country during this time before returning to Spain. This would include the UK, Ireland as well as Andorra.
Now if you spent only three months in the Schengen during the six-month period 14 March to 14 September, the new six-month period would start on 15 September. You have two options.
The first option would be to leave the Schengen area on 14 September and return three months later, say 14 December which means you could stay in Spain until 14 March.
The second option would be to leave the Schengen area and tale advantage of the start of a new six-month period on 15 September.
You would need to stay out of the Schengen area for a complete day (15 September). You could then return on 16 September to start a fresh three months. You would then need to leave by 16 December.
Usually, when you enter the Schengen area, you're asked to show you have made plans to leave the Schengen area within 3 months. I would therefore suggest you buy a return (two way) ticket to avoid any risk of being refused entry. You could get a ticket with a flexible return date, so you don't waste the return leg.
Otherwise, you'd need to buy a one-way ticket and then also buy your ticket for traveling out of the Schengen area (e.g. a flight to the UK or Ireland), so you can show you have made plans to leave the Schengen area by the end of the three months.
Finally, you mentioned you will be flying there with your cat. I assume your cat is in the US at present.
You will need to ensure that you have complied with veterinary regulations. This means getting a rabies vaccine for your cat and ensuring your cat is issued with a "pet passport" by your vet that shows when you cat was vaccinated. The full regulations are explained here:http://ec.europa.eu/food/animal/liveanimals/pets/nocomm_third_en.htm
I hope this answers your questions.