Welcome to Eorzea!

Unfortunately I don't do much crafting and I've never bothered with gathering so I can't really answer your first question, but for your second question about inventory space, as you play through the storyline around level 17 you would have joined the Scions of the Seventh Dawn, during which would have unlocked the Retainer system.

Although Retainers are specifically used for the market function, thus allowing you to sell items on the Market Board (MB), a secondary function Retainers have is that they also function as item storage for you, meaning you have somewhere to store stuff once your inventory gets too clogged (and it will). By default you can hire two retainers for free, but you can actually purchase more (up to three more I think) through the Mogstation website, which is an extra monthly charge to your standard subscription for the game. But this is completely optional.

Once you unlock retainers to hire them speak with a npc who has a title under their name of 'retainer vocate' (they're found in the market of each city, and are marked on the map with a special 'two faces' icon). When you hire a retainer, you get to choose their race, gender, facial appearance, name etc, just like when you created your character. Once this is done, to actually use them, find a summoning bell in each town or city and select 'Summon Retainer'. It's really simple once you get the hang of it.