A condottiero takes their name directly from the name of the contract (a condotta) that made their agreement. So yes, they would be bound by a contract, by definition.

As for finding clients, you should understand a little more of the social structures involved; the peerage of the nobility was international, it was a supra-regional fellowship, and even if the nobility fought each other, they also knew each other, socialized to an extent, corresponded, married into each other's families, travelled together, hosted one another, fostered each other's kids, etc. This same kind of extended social network was also present in the lives of the bourgeoisie, the urban merchant and craftworker classes, whose guilds were built on international trade connections. An up and coming guildsman might apprentice to a foreign master to learn a useful foreign language as well as the business at hand; the Nuremberger Michael Behaim, for instance, apprenticed in Venice for a time, and the famous Jacob Fugger also spent time in northern Italy learning the banking trade. Connections made by an apprentice could sometimes remain very close into their journeymanship and mastery.

A condottiero also makes use of these extensive international relationships, and famous captains came from the nobility and urban classes alike. Aristocrats, by their upbringing, were quite likely to be experienced in the practices of war, but as the centuries wore on the urban classes became more outwardly warlike as well, and it wouldn't be unexpected for a wealthy and powerful city politician to lead their men in a war, like when Wilibald Pirckheimer led the Nuremberg men in the Swiss War of 1499.

While both the urban elite and the nobility might have a martial upbringing and have access to a robust international social and professional network, the biggest advantage was their ability to access credit networks, and both the nobility and urban classes relied on moneylenders for liquidity when necessary. A condotta often stipulated an up-front payment, pay rates by the day or week or month, some arrangement for food and lodging, and a time of service, but all of these things were variable. The thing was that the clients needed men, fast, and so the social network of the condottiero was important; they call their cousins and their buddies and their old business associates, and they find men of the martial classes, who've had a bellicose upbringing and likely have their own armor and weapons already, and get as many of them together as possible, as fast as possible. The initial payment can be used as a sort of signing bonus for volunteers, and the rest can be used to purchase needed equipment - far more likely than buying weaponry, you'd be buying things like tents, clothing (or even just cloth to turn into clothing), wagons and wayns and draft horses, and any of the other stuff that sustains men on the march. Weapons like pikes or halberds, or crossbows or guns, might be purchased for men who don't have them already. It should be said that some of the more sought-after early mercenary forces had been sought precisely because they were already fairly well organized and drawn from bodies of men who already had that stuff. Genoese crossbowmen, for instance, were picked up because there was a large surplus of men willing to serve internationally in the manner in which they were expected to serve at home, so it's likely not many of them needed "training" in the same way that modern military recruits do. I've talked a but about who might join mercenary companies, and how those men might be armed, in previous posts.

Mercenaries, of course, have a pretty abominable reputation, and fiction that represents them tend to emphasize their inherently violent behavior, their untrustworthiness, their greed and avarice, but the thing is that this is how all armies were expected to behave. An army was a bad thing, full stop. It was full of wayward, violent men detached from their communities; community was a powerful way to direct those violent energies toward something less chaotic and murderous than an army. A community has systems of shaming and peer pressure and socially policed expectations of behavior, and an army breaks men free from those. So while at home these men probably wouldn't beat up a baker and steal his chickens, because that baker is their cousin Giuseppe whose mother taught them ribald rhymes, but in a different city, that baker is a guy with food, so who cares? Mercenaries were not an exception in either moral direction; violence, sexual violence, robbery, and arson were virtual certainties of any war fought by any group of men organized either as city hosts or mercenaries.

What we find when we look at examples of mercenaries breaking contracts is actually that the clients were the ones who failed to live up to their end of the bargain. We can understand a great deal of mercenary mutiny throughout history not as the inherent vice of uprooted men who fought for pay but a consequence of men whose boss violated their agreement. This was labor agitation in the same way that guild and even peasant rebellions were; the tactics were the same and ranged from work stoppages - for mercenaries, that might mean refusal to muster or march, or withdrawing from a siege, or refusing to carry an attack until they were paid - to outright physical violence. English mercenary archers under Charles the Bold mutinied in 1474 over unpaid wages, and even fired on Charles as he tried to negotiate with them. John Hawkwood's career is full of escapades that were attempts to extract promised pay by one means or another, up to sacking the city of Faenza. This is not to say that this was a good act, at all; Faenza was an especially cruel and brutal act, and it was rumored that Hawkwood himself murdered a nun in the course of it. It was a product of a very cruel military system that worked within a very cruel political and economic system that victimized the countryside as a matter of course. The men who did the victimizing can themselves be seen (and certainly saw themselves) as a victimized group in some ways as well. Please, again, understand that I am not saying that murdered civilians are in any way comparable to unpaid mercenaries, but to understand the mechanics of this system we should confront the fact that discipline in military hosts was as often a product of pay as it was a product of leadership or directed (or undirected) cruelty.

It's hard to find an example of a sustained military campaign any time in history that didn't involve troop mutinies or collective indiscipline over pay, and mercenaries were far from the exception. Switching sides was actually somewhat uncommon, but was usually a product of the client breaking a contract or failing to pay their troops, and the other side offering to pay them right now. Hawkwood was offered a very lucrative contract to switch sides after the aforementioned sack, but refused. Honesty was important, because a condottiero might only expect a few months of service at best, with frequent negotiations for renewals of contracts, and the somewhat bewildering politics of the period meant that a war could last decades or a just a few weeks, and making money as a hired man was never reliable; so your reputation of honesty and integrity was important. Men were careful not to run totally wild, but to do what we might consider heinous, hideous things in order to fulfill their end of the bargain for their men. Often that meant sacking and pillaging "friendly" territory, but breaking a contract was something seldom done without extensive and visible reasons.

So in essence, a mercenary is likely someone already connected to an international social network of armed and experienced men, has connections to creditors and those who can liquify assets for short-term cash flow, and were reliable, at least, to their notions of fairness in the contract, but operated within a military system of astonishing cruelty. The most visible actions of infamous mercenaries, like John Hawkwood, likely have their root in failure of the client to furnish promised pay as they do in any inherent violence of contract soldiery.

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John A. Lynn, Women, Armies, and Warfare in Early Modern Europe

David Parrott, The Business of War

J.R. Hale, War and Society in Renaissance Europe

William Caferro, John Hawkwood: An English Mercenary in Fourteenth-Century Italy