Do locked gauntlets prevent casting all divine spells as well as mysterious spells?
The locked gauntlet does not have an effect on the chance of spells spell Arcane spells. Therefore, the only problem that arises is locked gauntlets and the casting of spells concerns the somatic requirement. As it is indicated in the description of the locked Gauntlet:
As long as the glove is locked, you can not use the hand that carries it to cast spells or use skills. (You can still cast spells with somatic components, provided your other hand is free.)
However, you only need one hand to cast a spell with somatic requirements, as indicated by the rules:
A somatic component is a measured and precise movement of the hand. You must have at least one free hand to provide a somatic component.
Therefore, if you only use one locked gauntlet, you will have no problem casting spells. If you use two locked gauntlets, both of your hands will be used and you will need the metamagic spell spell or at least a third hand to cast spells with somatic requirements.
Do they prevent extraordinary abilities, magical or supernatural?
The use of locked gauntlets does not interfere with spell abilities.
Armor does not affect the use of a spell-like ability, even though this ability resembles an arcane spell with a somatic component.
The rules on extraordinary and supernatural abilities do not mention anything specific about wearing an armor or gauntlet. If there is an interaction between them, this will be specified in the description of the capacity concerned.
What skills prevent them? Only those based on strength and dexterity?
What happens if they do not involve the use of your hands (such as using acrobatics to jump over an empty space)?
The rules do not specify the number of free hands or the part of your body that you can use for a given skill test. All this part depends on GM's arbitrage and your imagination.
However, do not forget that the weight (5 lb) of the blocked glove applies depending on your armor, which can affect your skills depending on your load capacity and clutter effects. :
The indicated weight applies only if you wear a cuirass, light armor or no armor. Otherwise, the locked glove replaces a glove already present in the armor.
Can channeling energy (which needs to present one's divine purpose) still be launched?
The fact of being able present a divine home is rather ambiguous. Most GMs consider that a free hand is needed to do this when it comes to a symbol or a jewel, in which case it would not be possible with constrained hands. It remains more ambiguous when it's something that can not be directly selectedas with an exposed birthmark for example. Such cases would again fall under GM's judgment.