Is it possible to do a factory reset of iPhone 5 and keep iOS 6? I have an iPhone 5 that is currently running iOS 6. I do not want to update the phone to iOS 8, but would like to back up the phone's texts, reset the phone, and restore the texts onto the phone.

I don't want to restore the phone from a full backup since I only want to restore texts and contacts, nothing else on the phone. Is it possible to use iTunes to do this without updating the operating system and if so is there anything I should watch out for? Otherwise, is there some other third party application that I could use instead?
You cannot reset your phone, or it will need you to change to latest iOS 8. Apple doesn't sign old iOS versions which means it's impossible to restore an old iOS version.  
If you want to just wipe the texts and contacts, that's an easier task. 


If you have ever synced your iPhone with iTunes, you just need to sync it in the following fashion:  


if you usually sync the contacts and texts, you could uncheck the contacts and texts sync and erase those manually.
if you don't usually use iTunes, or need a faster approach to delete everything, I'd recommend using iTools PC/Mac app. (Just Google it).


Finally, you should use a blank space wipe tool (such as Battery Doctor) to assure that no erased data remains on the device.