What is the application process for a Non-O visa in Savannakhet, Laos and how does it differ from other locations?

If you want to get a 1 year multiple entry visa, get a NON OA. Number 6.

Sadly you (and everyone else) is out of luck trying to get a year-long, multi-entry Non-O visa (for any reason).. Since all the thai consulates in the world went to the eVisa online system they no longer offer the multi-entry choice (even though it might still show up in outdated embassy pricing sheets at their website) it is not a selection you can make.

They only offer 90 day Non-O visas now.. The way it works is you get the 90 day Non-O visa in your country before you come here (for what ever reason you're applying for it). You show up, get stamped in for 90 days, you get your thai bank account opened in your name only, you transfer in the required funds and 2 months later usually when you have 30 days or less left on your 90 day stap you apply for a year extension at the immigration office for 1900baht by meeting the in country requirements based on the reason you apply for the extension.

Then once the extension is inked into your passport you can buy a re-entry permit (single - 1000baht, multiple - 3800baht)

That's the way you do it nowadays :) Best of luck with it