Is there a way to gain an arcane discovery through feats, similarly to how one can gain a sorcerer bloodline powers?
Not in the RAW. If you can convince your GM to houserule, then sure.
1) No. The closest you can get is eldritch knight (if you already have at least one wizard level).
Diverse Training: An eldritch knight adds his level to any levels of fighter he might have for the purpose of meeting the prerequisites for feats (if he has no fighter levels, treat his eldritch knight levels as levels of fighter). He also adds his level to any levels in an arcane spellcasting class for the purpose of meeting the prerequisites for feats.
Most arcane discoveries have a prerequisite of minimum wizard level. There are a few that use the term Wizards spell. Other classes don’t cast wizard spells so those also cannot be used unless you are a wizard. Sorcerers have the same spell list of as a wizard but still cast sorcerer spells not wizard spells. There are 7 arcane discoveries that do not specifically call out a wizard level or reference wizard spells. Of those two of them affect your Arcane Bond. There are archetypes that grant arcane bond to other classes, but often the archetype specifies what the item has to be.
The list of Arcane discoveries that don’t specifically state a wizard requirement or affect the arcane bond are arcane builder, creative destruction, defensive feedback, and knowledge is power. Arcane builder is from ultimate magic the other are from Champions of Balance or People of the River. Bonded mask is from mummy’s mask and Ioun bond is from people of the river.
Balanced summoning seems to be a waste of time. You can already summon multiple creatures from a lower-level list. Being able to have two opposing templates does not really do much. Creative destruction, defensive feedback and knowledge is power don’t seem to be all that useful. Arcane builder could be useful, but crafting is really not that good unless the campaign is really limited on magic items. So, what arcane discovery are you wanting to take on a non-wizard character.
Most arcane discoveries have a prerequisite of minimum wizard level.
Irrelevant for VMC; because if that system wouldn't let you replace class level from the VMC's class with "character level", the system stops working. Virtually all Rage Powers say "the barbarian", for example.
About the same is true for when you gain a class feature from another class(usually from an archetype, but not always, cf. Skald).
The VMC will work but that is the only thing that will, but even then, some of them may not function properly anyways. Opposition Research is worthless to a cleric with VMC wizard.
Arcanist can get arcane discoveries via an exploit.
Arcanists can get ONE arcane discovery through an exploit. Choose carefully.
I mean most discoveries are very specific so you rarely want multiple.
I am of the rare group that wants multiple. I'd have been quite happy if the arcanist exploit that allowed for arcane discoveries was either unlimited (but capped effective wizard level to 1/2 arcanist level) or allowed you to select it no more than once every 5 levels. Or both.