Lots of questions. Ill add a little knowledge.

So first, a misconception. You started in Ul Dah as a pugilist. The thaumaturge guild is in that town, and that is the class that grows into the Black Mage job - certainly a DPS. And even if the MSQ hasnt taken you there yet, as someone who started in Ul Dah its possible to walk to Gridania(or vice versa). You pass through some mildly dangerous territory, but its doable at a surprisingly low level if youre cautious. Central Thanalan to eastern, cross the big bridge then continue north into the south shroud, from there to central shroud and to Gridania. Limsa is on an island, and the boat unlocks via the same quest as the airship so no early trips there. Gridania has the archer and lancer guilds(along with conjurer, leatherworker, carpenter and botanist).

Mining is a source of materials for crafting, or a source of income. You put in time and a little knowledge(knowing what to gather/where), you get out stacks of things to sell or fiddle with. Once you get to Limsa mining will certainly supply a lot of the materials that a blacksmith or armorsmith would use.

And i hate to be the bearer of bad news, but those level 1 mats arent likely to sell for much at all. Almost all of them can be found on the crafting vendors. Its around level 15 that mats stop being regularly sold by NPCs(players are still wont to sell anything and everything of course). As others have said, around level 17 in the MSQ youll be given the option to hire a retainer, and that NPC is how you sell things on the market.

So all your classes level separately. When you unlock ROG, you would be still be a level 21 PUG(or whatever level you are at that point) and a level 1 ROG. When playing as a ROG you use ROG skills. You can switch back and forth fairly freely(just cant be in a dungeon or during a fight). If you like ROG more you would eventually(probly fairly quickly) pass your PUG level altogether. Since theyre both melee DPS they share a set of "Role Skills", but everything else will be unique to one or the other. Rogue weapons(daggers) are mostly made by blacksmiths. And as for a second character, your first character can do and be almost everything so alts arent really required for much of anything. With that said, if you decide to make a ROG alt you should know that ROG is a special class. You cant start as one. You have to play something else for 10 levels(or more) and then switch. So if you wanted to do that, marauder would probly be the best bet as a starting class(arcanist also works).

As for skipping quests, you can. Lots of people will leave extra side-quests behind to use them to level up alternative classes later on. There are 3 types of quests to never skip. These are Main Scenario Quests(the meteor icon ones) - the quests that unlock most of the games features(if you skip these you literally cant do most things or go many places). Class quests - the ones that unlock some of your strongest and best combat moves. And unlock quests - optional quests that unlock more optional parts of the game(because you can never have too many options of what to do next - these will include everything from extra dungeons to changing your hairstyle to a trip to the Gold Saucer).

Good luck!