It's possible, but doing so won't provide the continuous barrier that helps keep moisture from getting through the concrete from the outside and into the finished wall materials, or to the concrete from the inside, possibly wetting the concrete or framing.

It's not a question of code, but more so a question of whether the insulation system you are construction will work effectively.

Depends on how much room you have to maneuver. I'm thinking you might be able to take a stud out in the middle of the wall to fit the insulation in, then slide them down the wall in both directions.

Plan B might just simply be to have closed cell foam sprayed.

As brewbeer points out, the big issue from a practical perspective is moisture control. The Building Sciences recommended way to manage moisture is by putting 1.5-2 inch XPS foam between the concrete and the stud wall. Putting foam in between studs won't keep moisture from hitting the stud or going between the "transition" between stud and foam. You should also be careful about putting fiberglass insulation in contact with concrete. Again, the XPS foam prevents direct contact.

Are your studs treated? In my county, by code you are not allowed to have ANY untreated wood in contact with concrete. So, XPS foam also provided a guaranteed separation between concrete and wood.

If you plan on having this intallation done "by code", you better find out what the code is for your specific area.

The "moisture" control is probably very dependent on location. Perhaps your particular setup doesn't have a lot of moisture in either direction. But, I would definitely want to know that for sure before going against the latest recommended building practices.