So, the most straight forward answer to most of your questions is that it varies - sorry, I know that's not helpful.
Most of these sorts of things are setting dependent, or are deliberately pitched in a way that is setting agnostic; Paladins may be part of an order that has a headquarters in a particular location ,if that's the way it works in particular world space or setting... or they may not, if it doesn't, and there isn't a hard and fast rule here.
In terms of strictly FR lore, and speaking primarily of the Sword Coast in particular, various religious and military orders exist, and could feasibly have paladins within their ranks, but you'd probably need to go to the actual novel series for anything that delves into the details of any particular order. Individual deities employ paladins directly as well on a one-to-one basis, but again, this isn't a rule, just another way that it *can* be.
As to why Paladins choose adventurous lifestyles... it's mainly because if they didn't, they would probably not be Paladins (in the adventurer sense); they'd be priests or acolytes, or religious scholars, or law-keepers or guards etc., to be an adventuring paladin is by definition to be of that out-ward facing, active way of living and acting in the world. Some military or religious orders may have members who are of the military rank of Paladin, but are not actually Paladins in the sense that we know them as adventurers; to be this type of paladin as found in the handbook, for playing this game, you are almost by definition one who will be living actively in the world - either by choice, directive, or by being thrust into that situation.
The consequences of oathbreaking are as many and varied as the reasons and motives that may lead paladins to break or forsake their oaths; exactly what the consequences are will be unique and individual, and there simply cannot be a one-size-fits-all rule for what happens. There are too many gods, too many divine purposes ,too many principles and ideals, and too many individual motives within each of those for a single rule of 'consequence' to be made. In game terms, it's very much in the realm of Dm's discretion. Losing your divine connection is a possibility, definitely, but it will depend heavily on the situation. In game terms, a DM might ask a Paladin player to switch to the "Oathbreaker Paladin" subclass (in the DMG for exactly this reason), gaining its benefits and losing the benefits of their original oath... but again, that's just one possibility.
How an individual paladin might react to an oathbreaker will be as individual as the person; some may seek to redeem them, some may condemn them, some my sympathise with their struggle and choice, some may be inspired to cast off their own oath as well ,after hearing the other's story... people are people, and their reactions will be unique... but as for whether they'd know on sight, generally not unless it's advertised or they have inside information - such as a warning from their own deity.
For the last, others may disagree, but I would certainly think so. Many people in the realms; most in fact, follow and give thanks to a variety of deities... few of them are actual priests or sworn to service. They are no less followers of those deities, however. A paladin who swears to the principles of heroism doesn't give direct fealty to any one deity for their oath and powers... but it's very likely that they will still give regular thanks or tokens of worship to various other deities, or one in particular if they're so inclined, in a way that has little to do with their oath - although, in that case, it is just as likely that the deity they follow may attempt to sway the paladin to declare directly for them, if their principles align closely enough.
