Robinson, 21, who previously had two loan spells at the Lilywhites, turned down a new deal with Villa and joins for an undisclosed compensation fee.
Newcastle-born former Rotherham man Pringle, 26, signs for an undisclosed fee after one year with the Cottagers.
Pringle made nine league starts during a short loan spell at Ipswich in 2016.
Robinson told the club's website: "It was obviously a big decision for me leaving Aston Villa, I've been there since I was eight-years-old.
"But, if there was ever a club for me to settle down with and sign permanently with, it would be Preston North End."
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Sheikh Obeikan, who was an adviser to the royal cabinet, opposed moves to relax gender segregation.
The dismissal comes shortly after Sheikh Obeikan attacked plans by "influential people to corrupt Muslim society by trying to change the natural status of women".
Saudi officials did not give a reason for Sheikh Obeikan's departure.
His recent comments were taken to be an attack on tentative steps towards relaxing some stricter interpretations of Saudi law.
King Abdullah has promised women the right to vote in future elections, has opened the country's first co-educational university and introduced measures against domestic violence.
The head of Saudi Arabia's religious police, the "mutawa", was also replaced in January, with the new head widely seen as more moderate than his predecessor.
Sheikh Obeikan achieved a degree of notoriety several years ago with a decree suggesting that unrelated Saudi men and women could mix so long as the man drank the woman's breast milk, thus creating a maternal bond between them.

After James Tavernier was penalised for a last-minute challenge on Jonny Hayes, James Maddison curled home a fantastic set-piece from 25 yards.
"The referee handled a difficult game very well," said Warburton.
"He did a really good job up to that point, but to give that one there is frustrating beyond belief."
Warburton is convinced Tavernier made an "excellent tackle" and confronted referee John Beaton on the pitch at full-time.
"I will be careful what I say about the free-kick," he added. "Everyone saw the ball move. The fourth official on the sidelines saw the ball move and made a comment. And yet a free-kick is given.
"You have got to be, I think, 100% sure in these type of games, in those areas of the park.
"Late in the game, when your players are fatigued, I thought Tav made an excellent tackle there. I don't think their player moaned about it and it is a huge decision to give."
Rangers dominated the first half as the teams met for the first time since January 2012.
However, the visitors were caught cold in the opening minute of the second period when Hayes broke clear to score.
A surging run from Rangers captain Lee Wallace was halted by Hayes, leading to a penalty converted by Andy Halliday.
Maddison's stunning strike left Rangers in the bottom half of the table, with just two wins from their seven league outings, but Warburton insists his team are moving in the right direction.
He said: "We dominated the ball first half completely but we went in at half-time frustrated it finished 0-0.
"We gave away a schoolboy goal and for three or four minutes we looked a bit rocky. We lost our composure then got the ball down, then played. We got our rewards via the penalty and I thought there was only one team in it.
"I am never happy to drop a point, never mind three. But we moved to the next level as a team and we are getting better.
"The boys are settling in and we were delighted with long periods of that game where we completely dominated."

Gwent Police said it was "extremely concerned" after five incidents potentially linked to heroin were reported between 23 October and Friday.
They happened in Caerphilly, Newport and Blaenau Gwent.
Det Insp Leigh Mears said he "cannot rule out the possibility that the deaths and the medical emergencies are linked to the same batch of heroin".
He added: "The use of heroin is of course illegal, but I would urge anyone using the drug to be especially cautious and vigilant.
"If they experience any unusual symptoms after taking drugs, they should seek medical attention immediately."
Championship side Preston North End have signed Aston Villa forward Callum Robinson and Fulham midfielder Ben Pringle, both on three-year contracts.
[NEXT_CONCEPT]
Saudi King Abdullah has sacked one of his most hardline advisers, Sheikh Abdelmohsen al-Obeikan.
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Rangers boss Mark Warburton was furious with the award of a free-kick from which Aberdeen scored their winning goal in a 2-1 victory at Pittodrie.
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Police have issued a warning about heroin after the deaths of three people in five days.