I was told years ago from my Thai bank that any value over 10,000 USD is liable to a higher risk of investigation for money laundering, meaning legit transactions could be delayed, I’m not suggesting your payment has been delayed nor am I suggesting any illegality going on here but if you’ve made regular transfers above the 10k usd then maybe they’ve been monitoring your transactions and thus caused issues.
Usually they check, but I’ve been asked once when using a one way ticket if I’d planned on doing an extension once there, (which I did ) this following the check in desk checking through my passport pages and I guess them seeing mostly yearly extensions or yearly multi entry visas with plenty in out stamps, so you may get away with it but best have a dummy onward ticket just to be sure.
Even with the paperwork I’d still leave the car at the border parking, savannakhet its easy to find a Tuktuk anywhere, you may have to stay at least overnight, but you can ask the immigration when you check out whether they will allow you to return early evening, be aware of fake booze, I brought back a bottle of gin from the nice duty free shop in town a couple years back for a friend, I was quite surprised that he finished up binning the bottle after pouring a glass.
There is unlikely to be any issue with your criminal convictions as you mentioned they are not serious, over 10 years convictions are often considered spent anyway depending on offense, I have been staying here on visa based on marriage, and also retirement visa and wasn’t asked for any police checks. Depending on the conviction some convictions can be removed in as little as 7 years and can also be requested to be removed particularly if you were under 18 at the time of offense, this how it used to work in Scotland anyway,