Is there any process or series of plays/cards that would allow an instant to be put onto the battlefield as a permanent, instead of resolving? Would it resolve anyway, even if not cast? Not including something like manifesting an instant, where it's not technically the card.
There's a clear boundary between the two types of cards (one having a 'lasting' effect, the other a 'momentary' effect), and as far as I can see that's intentional. In the history of Magic: The Gathering, a handful of instant/sorcery cards with transform and/or manifest have been printed, and with some other cards it would be possible to make permanents of them, except for some very specific rules (mentioned by @doppelgreener) preventing this:
304.4. Instants can’t enter the battlefield. If an instant would enter the battlefield, it remains in its previous zone instead.
307.4. Sorceries can’t enter the battlefield. If a sorcery would enter the battlefield, it remains in its previous zone instead.
701.27d If a spell or ability instructs a player to transform a permanent, and the face that permanent would transform into is represented by an instant or sorcery card face, nothing happens.
701.33f If a manifested permanent that’s represented by an instant or sorcery card would turn face up, its controller reveals it and leaves it face down. Abilities that trigger whenever a permanent is turned face up won’t trigger.
Even if those rules didn't exist, the text on an instant or sorcery indicates what happens when the spell resolves. So if you could somehow get a Lightning Bolt on the battlefield as a permanent, it won't do any damage, not even when it enters the battlefield.
It's not possible, but it almost was.
In 2001, Wizards added this bit of errata:
They don't say what the interaction actually is, but there's a good chance that it was related to the card Retraced Image (which would be released in Torment about 6 months later), which read
Reveal a card in your hand, then put that card into play if it has the same name as a permanent in play.
In the absence of rules 304.4 and 307.4, this would allow you to put the (Sorcery) card Splinter into play if you had a Splinter token in play. Which is, as they say, "an unintended and strange interaction."
I have not been able to find any evidence as to whether or not rules 304.4 and 307.4 (which put the kibosh on these sorts of shenanigans) existed at the time. They certainly existed in 2006, when Time Spiral reprinted Splintering Wind and reerrataed the tokens to be Splinters again.
Sort of, but not really.
As Glorfindel said, not currently. However, this card was very recently shown at a MtG event where packs contained experimental/prototype cards. It does exactly what you're looking for.
Speculation: If they're testing out gameplay of this mechanic, they may be considering releasing something officially that does something like that.
This is not possible under normal gameplay:
400.4a If an instant or sorcery card would enter the battlefield, it remains in its previous zone.
Of course, silver-bordered or other non-tournament-legal cards (including the Mystery Booster test cards) may bend the rules in ways not covered by the Comprehensive Rules. In such cases, absent official rulings, this would be up to your playgroup to decide. (The rulings for the test cards can be found in the Mystery Booster release notes.)