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Is anything shared between characters for crafting? Should I be doing research on every character or just one? Just trying to figure out what the benefits are to crafting on my main or just making a crafting alt.
Alchemy and provisioning are also shared at least in part. This means you can craft potions & poisons or food and drinks on one character, then put them in your bank and other characters can take the things from the bank and use them for daily writs on a different character. For us lucky people on PC, Dolgubon's Lazy Writ Crafter can even take the required things from the bank automatically when you interact with the bank.
It's definitely a good way to avoid cluttering all your characters' inventories with the different products (there are many alchemy products which are needed in writs!). I keep these things in my bank and pre-craft two stacks or so when I run out of a particular item.
Research on alts will allow you to transmute/reconstruct gear in those traits on those alts (rather than changing back over to your main)
Research also increases the master writ drop chance for BS/WW/Cloth/Jewel writs when done at maximum level.
Motif knowledge increases the master writ drop chance for BS/WW/Cloth
Traits "learned" for Alchemy and Enchanting increase the master writ drop chance
Number of Purple/Gold recipes known increases it for provisioning.
These are all character based, so to increase master writ chances you'd want to do those on those characters.
Thankfully, if that doesn't interest you, ESO allows you to share (most) gear among all your characters, so you can get by with just one dedicated crafter.
You can do trait researches on Alts if you're short on bank space holding up future items for research on your main. By doing it on an alt, you can recraft that item at a later time when your main has a research spot open. It's 1 less bank space used up per item you're holding on while you wait for researches to complete.
Is anything shared between characters for crafting? Should I be doing research on every character or just one? Just trying to figure out what the benefits are to crafting on my main or just making a crafting alt.
Biggest one is skill points. It's much easier to get enough skill points with a separate crafter and scryer - you level to 45 in random normal dungeons farming crystals, and/or battlegrounds farming AP then use the skill respec freebie scrolls to get all the skill points back to use them as a crafter or scryer because at 45 you can get DLC dungeons so they cease to be good crystal farmers. Battlegrounds is a good one to consider for levelling a crafting toon especially during double AP events where the under 50 BG is full of non PVP players and you'll level to 50 with about a million AP which lets you buy a load of the crafting motifs sold for AP instead.
Second one is CP. Even ignoring the green tree slotting mess - which will presumably get fixed next update or two - you get to use the same CP on every toon (even level 3 toons who've walked out of the tutorial into a crafting spot). That means that if you don't have 3000 CP you can still put meticulous disassembly on your crafter, treasure hunter on your adventuring toons and the various farming speedups on other toons.
For non-crafters you are levelling it is useful to learn the essential traits for your class/build early (eg divines or training) you can then reconstruct armour sets in them as you level. Once you've bound a full set it's 25 crystals to make, 25 to decon a piece so levelling gear essentially becomes free. Need to level up your mother's sorrow, decon, recon, done. Need to repair your green training armour, decon, recon instead. Learning two traits in everything you use for that toon is pretty fast as well.
As an aside - another reason to level up one of the wood/blacksmith/clothing skills on some toons is to get a regular supply of high level gear repair kits so you never have to get ripped off by the merchants again.