Both ways tell the same. Los teléfonos son públicos doesn't mean that they are THE public or people, but they are destined for the public. For instance, for a subway (el metro), you can say El metro es público. Público, apart from a noun also works as an adjective, i.e. la escuela pública (state school).
Early on in learning a new language it can all feel a little haphazard, but this one is pretty straight forward. Telephone is a cognate in Spanish, teléfono, which is masculine as you would expect from a noun ending in o. So the plural is Los teléfonos. Since you are talking about an essential characteristic of the telephones, the correct verb to be is ser. The telephones can be replaced by they and in Spanish Los teléfonos could be replaced by Ellos. The third person plural form of ser is son. Público here is a predicate adjective modifying teléfonos so it is also in its masculine plural form públicos.
Los teléfonos son públicos.
That would be Los teléfonos son para uso público. Every small change like that you make has a similar small change in Spanish. Part of understanding a new language is knowing what is said and what is not. When you say that The telephones are public that is assumed to mean for public use, but translation in these exercises should be more precise except when there are differences in the mode of expression in the two languages.
Absolutely. The accent marks are one of the greatest features of Spanish. Essentially Spanish has two basic patterns for determining which syllable is stressed. The first use of the the accent is to show when the accent is on a different syllable than the rules would dictate. The rules are:
If the word ends in a vowel, an n or an a, the accent is in the penultimate (next to the last) syllable.
If the word ends in any other consonant, the stress is on the last syllable.
So this explains these two accents here. Without the accents, these words would be pronounced teleFOos and pubLIcos respectively instead of teLEfonos and PUBlicos.
There is a similar use where the accent breaks up a dipthong. You will see that mostly at the end of words with the ia or ía endings. You can hear the difference between words like historia and alegría.
The other use of the accent has nothing to do with pronunciation. These just distinguish two forms which otherwise would look like another word. You probably already recognize most of the one syllable ones like el the vs él or si if vs sí yes. What many people don't recognize is that all interrogatives (quí, qué, cuándo, dónde, cómo and cuál) have accents because there is a different part of speech, often a relative pronoun, which looks the same but has no accent. This is easy with qué and que because they translate differently. But some of the others like dónde can be more difficult. A sentence like There is no internet where I am staying would use the form without the tilde. No hay internet donde me quedo.
I hope that helps.