What does being a flag carrier mean for an airline?
It's a bit of an obsolete term now, in terms of practicality, but it basically means a national carrier. Historically this had a lot of legal implications about who was allowed to fly international routes, and also historically these were almost universally actually state-owned and got a massive amount of special treatment, financially speaking, relative to other airlines.
Now a lot of "flag carriers" are privately owned (Korean Air and British Airways are two examples) and plenty are privately owned subsidiaries of the same company, so the definition is pretty loose - it just means a carrier heavily associated with their country's national identity, and essentially all formerly state-owned carriers are grandfathered into that. The legal sense is looser too - basically any airline can apply to operate international flights. They still get special treatment financially a lot though.
Some countries have multiple flag carriers, and some countries share one, probably most notably Sweden, Norway, and Denmark making up SAS. More or less the only exception is the USA, which has never had an official flag carrier.
In terms of liveries, it means absolutely nothing, but I do tend to find it slightly more acceptable if your color scheme is heinous if it's because it's the flag of the country you represent. I don't think it's a requirement or anything, just a mitigating factor.