Not 'capisce', which is pronounced cah-pee-she, but 'capisc', which is pronounced cah-peesh. Dialect from Naples frequently drops the last syllable. Similarly 'ricotta' becomes 'ricot'. Note the accent is where it would be if the last syllable were still there: cah-PEESH(-e), ree-COHT(-tah).
Someone needs to watch more episodes of The Sopranos! ;-)
I disagree. you can use both forms to directly speak to a person. the way you address them is different though. 2nd person singular is with "tu" and third form is with "il signore / la signora..." although you can spare the noun. so capisce isreferring to il signore / la signora and capisci to tu.
Yes, it is based on context. Almost always in Italian, the definite article (the) is placed before a noun (there are a few exceptions). For example if you said "I want to go to university" in Italian, it's literal translation into English would be "I want to go to THE university". This can be annoying in Italian since it means he might understand women as a species or there might be a group of French women that have ended up in China and nobody else can speak French but he can so he understands them. Who knows?