I'm wondering if these skills have any story applications beyond their basic functions.

For example, are there things out and about in the world that my character could find and repair for a bit of story or allow entrance into some extra location in a dungeon, or is it just for keeping my gear/mechanical companions in good condition?

Is Gambling just for earning regular/high-quality inventory items or can I get an NPC to bet a quest/story-related item (depending on the circumstance/if they're a gambling person, etc)?

Is Pick Pocketing useful for more than lifting extra money and items off people or are their instances when I could steal a quest/story-related item off an important NPC?

=) I'm not looking for specifics, I just want to know if these situations exist in some form. Thank you muchly.

Yes, there are definitely instances where you can pickpocket quest or story items. Pickpocketing is often a viable alternate way to solve a quest. I don't know about repair and gamble though.

There's one Gamble quest (pretty cool, but it just advances the main plot).
No Pick Pockets quests (and getting mastery involves doing something that will reduce your rep), but plenty of quests that have a more expedient (but not more rewarding in terms of game content seen) Pock Pocket solution.

The Thieves' Guild (if you become a member - there are two quests that, separately, can land you an invite, and one quest for membership, all three perfectly solvable without killing and any thievery skills at all) offers a bunch of "go there, steal X, hand it over for XP and money" quests. These ultimately lead nowhere, except you can hold onto the good items - but the good items are perfectly stealable without quests or membership (or even thievery skills).

Gamble and Pick Pockets are useful to get things that people wouldn't sell (ring of persuasion, various guns, a must-have dress for the female PC) - but there are always 1) Fate Points, force critical success option and 2) Baneful Gauntlets, drop them near the npc and see what happens, hehe.

Repair is not worth it. Don't get your stuff damaged. When fighting a bunch of armor-damaging monsters, put a healer on standby, get the monsters to follow you and run in circles while your other party member with an unbreakable weapon kills the monsters one by one (since the monsters in question are unintelligent undead and elementals, I don't count it as an exploit, just good roleplaying).

The final consideration when choosing skills are the content of the mastery quests. The quests with the most plot are Melee+Dodge (together) and Persuasion, the ones with plot impact (different game endings) are Persuasion (again) and Repair (available without the skill itself).