iOS 10.0.3 just like all Apple iOS 10 releases, should be compatible with the iPhone 5 or later, iPad 2 or later, 6th generation iPod touch or later, iPad mini 2 or later and iPad Pro range. However, Apple has favored to release it only for the iPhone 7 and iPhone 7 Plus.
Getting iOS 10.0.3 should be simple. Upgrade prompts should be received automatically. But if that doesn't happen in your case, you need to trigger it manually. Don't worry, just go to Settings > General > Software Update.
Finally, jailbreaking is not an issue with a new iOS 10.0.3 release because it's only for the iPhone 7 and iPhone 7 Plus that is launched with iOS 10.
When we are talking about bugs, so far Apple's official Support Communities forum is still pretty quiet about isolated reports of Wallet and Voicemail bugs, and regarding Touch ID becoming unresponsive. The bad news is that cellular connectivity problems are spread through multiple iPhone models and carriers and iOS 10.0.3 seems to be working well only for the iPhone 7 and iPhone 7 Plus (as we already said).
Information about new iOS 10 is thin on the ground. We don't know detailed release nor information about security content whatsoever. In fact, the only thing we can go with is vague description: "iOS 10.0.3 fixes bugs including an issue where some users could temporarily lose cellular connectivity."
So far, there are many questions on whether iOS 10.0.3 satisfy its primary criteria. It remains to be seen but right now at least it isn't bringing in any new problems.
Also, there is a possibility Apple will release iOS 10.0.4 given the cellular connectivity issue is proving to be more far-flung than the relent of iOS 10.0.3. Having that said, Apple might choose to roll a wider repair into the upcoming iOS 10.1 that also introduces a great new camera mode for iPhone 7 Plus and new modem firmware - is due out this November.