You know in a settings like one piece how the hell did they execute the pirate king in a town which at least in the present doesn't seem to have an of the really strong people. I mean with his title and leaving a legendary treasure he is supposedly pretty high on he power scale and what looks like normal small fry soldiers are escorting him. Maybe his wooden cuffs contain seastone on the inside but just turn on emperor's haki and make the whole crowd faint.

I don't know how Luke/Gorden are going play it out, but in canon that question eventually get raised by the Straw Hats when talking with one of Roger's old crew members.
It's revealed that Roger had long been suffering from an incurable disease and knew his time was near. He choose to voluntarily give himself up to the Marines and let himself be executed, probably for the express purpose of ensuring a world-wide audience for his "My treasure's yours, if you can find it" speech.

He trolled the world gov big time on that one.
Now i wonder, what are theses guard? Which corps do they belong? they look neither like Marines or agent and i think this is the only time we see soldiers in Kaki colored outfit in OP.
Are they from another governement force?
Are they excutionners? or maybe they are specific from loguetown.

Twist: Those ordinary looking guard are actually highly competent member of a excutionners corps. all of them possess a high level of mastery of the armament and perception haki and competence in every technique of Rokushiki, and their weapon are made with seastone, too.

He may have let it happen. I admit I don't watch the show so I don't know if there is a canon reason, but he may have let himself be killed willingly. Like, to protect family, as part of a deal, to do the ultimate spite to the marines by kicking off a grand age of piracy, or something else.

It was actually All of those. He was dying of an Illness anyway so he concocted a scheme where he could force his greatest Enemy (luffy's grandfather) into promising to look after his unborn son while simultaneously giving a big F*$^ you to the marines by kicking off the age of piracy.

It's also possible that stronger people chased him down and cuaght him, dragging him to the nearest town to then do the deed. Or maybe he bargained to be offed in the East Blue, where he new his words would be most effective (his final speech would spread to the stronger seas, no question, so he wanted to make sure that everyone heard. That meant starting with the weakest.) But that's just a guess.

If I had to guess, he requested that he be executed in his home town, and as he turned himself in he was in a position to make such a request.

Ideally, he would of turned himself in or nearby the place.

no-one going to comment on the "spears?"