Do locked gauntlets prevent casting all divine spells as well as arcane ones?
The locked gauntlet has no effect on the Arcane Spell Failure Chance. Consequently, the issues there are with locked gauntlets involve the need of a free hand. As it is stated in the Locked Gauntlet description :
While the gauntlet is locked, you can’t use the hand wearing it for casting spells or employing skills. (You can still cast spells with somatic components, provided that your other hand is free.)
A free hand is required for somatic components, using material components and manipulating a divine focus.
However, you only need one hand to cast a spell with somatic requirements, as states the rules :
A somatic component is a measured and precise movement of the hand. You must have at least one hand free to provide a somatic component.
Manipulating material components and divine focus have the same free hand requirement. Note that the same free hand can perform the three actions when you need to cast a spell with the three components.
Therefore, if you use only one locked gauntlet, you won't have any issue for casting spells. If you use two locked gauntlets, both your hands will be used and you will need some ways to get around the problem.
The Still Spell metamagic feat allows you to bypass the somatic component issue. The Eschew Material will do the same for the material components issue, when they cost less than 1 gp. Finally, a reliquary weapon, armor or shield can be used as a divine focus, solving this issue. Having at least a third hand also allows to bypass all these issues, but it maybe isn't the easiest solution.
Do they prevent extraordinary, spell-like or supernatural abilities?
Spell-like abilities are not impeded by the use of locked-gauntlets :
Armor never affects a spell-like ability’s use, even if the ability resembles an arcane spell with a somatic component.
The extraordinary and supernatural abilities rules don't mention anything specific about wearing armor or gauntlets. If there is any interaction between them, it would be specified in the concerned ability description.
Which skills do they prevent? Only strength and dexterity based ones?
What if they don't involve using your hands (like using Acrobatics to jump across a gap)?
Rules don't specify anithing about how many hands free or which part of your body you can use for a given skill check. All this part is dependant on the GM adjudication and your imagination.
However, don't forget that the weight (5 lbs) of the locked-gauntlet apply depending on your armor, which can affect your skills depending on your carrying capacity and the encumbrance effects :
The weight given applies only if you’re wearing a breastplate, light armor, or no armor. Otherwise, the locked gauntlet replaces a gauntlet you already have as part of the armor.
Can channeling energy (which requires to present one's divine focus) still be casted?
The fact to be able to present a divine focus is rather ambiguous. Most GM consider that it requires a free hand to do it when it's something like a symbol or jewelry, in which case it would not be possible with constrained hands. It remains more ambiguous when it is something which cannot be directly picked, as with an exposed Birthmark for example. Such cases would once more fall under the GM adjudication. As for the divine focus requirement for spells, a reliquary item should however solve the issue.