Public toilets are not widespread and those that do exist are often closed; some make a small charge. The best thing to do is to nip into a cafe or bar, all of which are required by law to have a toilet.
Rome is a safe city but petty theft can be a problem and pickpockets are active in touristy areas and on crowded public transport. Use common sense and watch your valuables.
There aren't any no-go areas, but keep your wits about you around Stazione Termini.
August is Italy's main holiday...