Alright, so the short answer to everything you've asked is "yes." But that's boring and not very helpful, so here we go!

There were no laws in any state that I'm aware of in antiquity prohibiting the ownership of weapons generally. A few states limited the number and type of weapons, and some (like Lacedaemon with her helots) prohibited certain segments of the population from carrying them. But citizens, and in most places non-citizens (or at least metics) could legally carry arms. Generally weapons were small and kept hidden, however, since the open display of a weapon was bad manners and large weapons were unnecessary for self-protection and just annoying. Of course, if you actually used your weapon there were usually consequences, the most usual being that a guard would probably come and beat you up if he caught you. But of course, in the case of murder...

As can be imagined, small handy weapons would be most common. People didn't go running around with spears and swords usually, unless they were soldiers. Daggers, knives, clubs--there's actually a sophisticated vocabulary, particularly in Latin, that enables you to identify various types of small weapons, although those words are rarely used.

Your third question is the big one and the important one. Yes. Yes. Yes. What we would characterize as gang warfare was extremely important in ancient politics, particularly in Rome, but also in Athens. The Athenian general Alcibiades was famously implicated in the impious disfigurement of the herms on the eve of the Sicilian Expedition because he was known to be one of a group of drunken aristocrats who often participated in theft and vandalism after a night of partying--essentially a gang, although usually not violent. Street thugs were also famous, particularly at Piraeus, were the streets were thick with crime.

But Roman gangs are more important. There was recently a similar question about them, but I've misplaced it. So there were plenty of thugs and gangs of bandits running around the city--so much so that Augustus had to set up various organizations of guards and watchmen to protect ordinary citizens from them. But they aren't the interesting or important ones. That honor goes to the political gangs.

See, the way that Roman politics worked was that aristocratic or wealthy candidates would build up client-bases for themselves, bands of loyal followers and/or allies. These tended to be ordinary, law-abiding citizens or important political players, but they weren't always. Frequently the clients of one candidate or party or another would field what were essentially street gangs. These gangs would clash in bloody street battles with each other, somewhat similar to what we might classify as riots today, often necessitating the intervention of the military. These street gangs were of incredibly great importance, swinging people's votes rather like the way the mafia might today.

Most important of the political gangs known to us are those of Milo and Clodius. Clodius Pulcher is often described in introductory materials as a demagogue, but that's not really what he was. Clodius was a rabble-rouser, ready to do whatever it took to propel himself forward in his career. Clodius was the leader of the largest gang in Rome, which owed nominal allegiance to Caesar and the remnants of the Marian party. In reality, Clodius was looking out for one guy: Clodius. And Caesar knew that perfectly well, which is why he kept Clodius on such a long leash. Clodius' rival was Milo, who was probably even more despicable than Clodius. While Clodius had actual political ambitions which took precedence, Milo was essentially just a thug, and his political agenda was definitely second after his willingness to gather followers and do damage to his opponents. The gangs of Clodius, nominally backed by the Caesarians and Pompey, and Milo, nominally backed by Cicero and the optimates, clashed violently several times. Things finally came to a head in the elections of 53, B.C. when Clodius was a candidate for the praetorship and Milo was up for the consulate. The outcome of the election would, at least for the time being, show which political group would end up in control of Roman politics for the next couple years. Clashes heated up throughout the year, until finally on January 18, 53, B.C. the rival gangs of Clodius and Milo, one way or another (the accounts vary), clashed once more with each other. The result was that Clodius and most of his entourage were killed. Well the result was pretty monumental, since it caused a massive riot at Rome that burned down the Curia, and subsequently Cicero defended Milo at his trial. Because of the speech that Cicero gave in Milo's defense, Milo was exiled to Massilia but escaped with his life, which was an outrage to Caesar and his following. The long-term result of this was that Pompey, who had kept the riot in check with his troops, was awarded an extraordinary magistracy, the consulate without peer, in order to restore order. Because of his successful coup Pompey shot up in Roman politics and was within grasp of his ultimate goal, total control of the Roman state. It pushed Caesar way into the background, and nobody expected him to bounce back so impressively.