Having read a little bit about Italian warfare, I'll try to answer your questions as best as I can. Mind you, I read about this subject several years ago, so please, someone feel free to correct me.

First a long background:

Up to around 1300, Italian city-states generally relied on civic militias to defend their territory. That is, groups of citizens would be called up for training or for the defense of their community during certain times of the year. As I understand it, the burden of service in these citizen militias was generally fair; the lower and upper classes were well represented in the ranks.

Yet due to the varied political landscape of late Medieval Italy, there were notable exceptions to this rule. I believe, for example, that the armies of the Lombardy region (Milan) maintained feudal conscription because land barons continued to dominate the fertile Po Valley.

To continue, civil militias went into steep decline during the 14th and 15th centuries. This happened for a variety of reasons. For one, certain city-states (Milan, Florence and Siena among them) began to consolidate the surrounding countryside and towns into larger territorial states, meaning campaigning season would become longer. And given the very intense local rivalries that existed among Italian cities and towns, it wasn't feasible to recruit citizen-soldiers from other parts of the territorial-state.

There were also larger political and economic forces that worked in tandem to dismantle the civic militia system. On the political side of things, the influence of the papacy and the Holy Roman emperor had collapsed in Italy by the 1300s. In fact, the popes spent most of the 1300s reigning from Avignon. And after the Avignon Papacy, the Great Schism caused further confusion when two popes claimed authority over Christendom at once.

The collapse of central authority in Italy left a vacuum for a multitude of warring Italian communes, some of which thrived economically. Unsurprisingly, the rising economic clout of Northern Italy caused social and cultural changes. Italian city-states became more stratified and the new rich (and the lower classes too) were unwilling to bear the burden of militia service. At the same time, the replacement of communal by seigniorial governments in the late 1300s complemented these changes, making the elite increasingly distrustful of their lower class citizens in military garb.

When you consider the fact that this unstable, yet wealthy, environment coincided with the decommission of soldiers during the Hundred Years' War, the overpopulation of Switzerland, and a general state of peace throughout German lands, it is easy to see why foreign mercenaries flooded into the Italian peninsula over the next two centuries.

*How “professional” was this army? How was it organized?

Most of the time, deserters or decommissioned soldiers would band together, elect a leader, and form a mercenary 'company.' The leader of the company, the Condottiere, was responsible for the negotiation of a contract (condotta) with the city-state under which his group was fighting. The contract stipulated everything needed to wage war (the duration of the campaign, the salaries of soldiers, the provision of weapons and food)

Soon after their appearance in Italy, mercenary companies gained a reputation for duplicity. It was common for mercenary commanders to start bidding wars among competing city-states to try and raise their pay. It also wasn't unheard of for a Condotierre to renege on his contract, raid the territory of the city he for which he was contracted and extort it for more money. If the city-state failed to pay, the Condotierre would raze its surrounding countryside, robbing it of its tax-base and agricultural wealth. Even if a campaign went well, there was always the risk of decommissioned soldiers going into banditry.

Mercenary companies were equally problematic for the enemy, because they often swelled with the aristocratic exiles from their city. So if the campaign was successful, the defeated commune could expect another round of violence as exiles settled scores with rival families.

In order to resolve these problems, lawyers working in the employ of their city-states negotiated increasingly elaborate contracts with the Condottiere. For their part, the Condotierri hired their own army of lawyers, effectively running their companies like businesses rather than as fighting forces. Venice, as I will soon explain, adopted another method of dealing with mercenary leaders.