Hi Russ, Yes, you should update to 10.3.1. As well as one or two bug fixes, there will be security updates.
You're welcome. You probably won't notice any great difference. I had no problems with my 6S -( 32 GB also) and yours is a newer device, which will be even more capable of updates.
I did it OTA with WiFi. It took about 25 minutes (it depends on your connection). I always plug my devices in when I do an update, unless it's very small one - Just make sure you have plenty of battery power. It's always a good idea to do a backup just to be on the safe side.
In the Videos app, if your iPhone is not up to date. If that app doesn't exist, look for a TV app. The movie should be there.
Your other post with the same question has been removed. Two threads asking for help with the same problem is confusing, and you'll receive more helpful answers if the replies just concentrate on one thread.
You're welcome. The Videos app has been replaced by Apple with TV, for users in the USA:
Apple unveils new TV app for Apple TV, iPhone and iPad
The latest version would be 10.3.1. As Caroline told you before, it comes with bug fixes and security updates. Keeping the software up to date on your iPhone is always a good idea.