GEICO is the low-cost provider for a reason; expect no frills. They try to build office camaraderie through other ways (charitable giving, award ceremonies), but it largely depends on your supervisor (and your supervisor will change several times during your tenure).
Expect process on process on process - if you can't keep up with emails letting you know something has changed or you can't roll with the punches, this may not be a good fit. Markdowns aren't always fair or applied judiciously.
Sometimes processes are strange and there's no explanation. Or processes are strange and don't make sense until you move to the next "level" of claims because you didn't yet have the knowledge to understand the rationale.
Recruiting is a bit disingenuous; for example, the MDP program is pitched as a fast-track to management. But management is really supervision and there are probably 90 supervisors in the building. Once at supervision, you then have to advance through 4 more levels of supervision to get to a manager role, yet the program was painted as a fast track to the top. I have no doubt they lure general claims with promises of best-case-scenario promotion timelines.
Eventually, if you get high enough in management, to progress further you will need to relocate. Not necessarily a con because GEICO is generous with relocation assistance if they want you and will long-term try to get you back to office of choice, just something to keep in mind.