You can find plenty of prior questions on this topic here, but I'd reccomend checking out the responses from /u/Brigantus and I at this recent thread first.

It is of course impossible to know whether or not our pronunications are exactly the same as they were 1000s of years ago. Odds are, they aren't exactly the same. But we're doing our darnedest to figure out what it was like back then. There's two main techniques we use:

• Extrapolation from descendants: languages for which we have ancient texts often have modern descendants with millions of speakers- Latin, Mayan, Chinese, etc. Additionally, languages evolve in regular ways. A linguistic community will alter the same sounds in one language in the same ways often enough to make a different language. For instance, Latin words with an initial "s-consonant" sound take an "e-" in Spanish but not in English. Latin statum is equivalent to English "state" and Spanish estado. Specialis developed into "special" and especial. Latin schola (orig. Greek) gives us "school" and escuela. We can extrapolate backwards from these and guess at the original sounds. For Ancient Egyptian, Coptic is the closest decidendant. It has not been nearly as commonly spoken as the aforementioned languages, and has survived primarily through Coptic Christian liturgical tradtion. But it's something.

• Comparison with other languages: The Rosetta stone is famous for good reason. The best thing we can hope to find when deducing the meaning of ancient phonetic signs is to find known words written using them. The Rosetta Stone contains the same text written in Demotic (script form of Egyptian), Egyptian heiroglyphs, and Greek. We know names like "Ptolemy," so when we see the same name appear in Egyptian, we can work out the meanings of the individual symbols. The same applies to other ancient scripts: we know the names of several Aztec deities from Spansih colonial sources, so when those names appear in earlier Mayan heiroglyphs, we can check our phonetic translations with them.

• Cross-referencing with literary and morphological convention: For finer details, we can often see how our translations fit into literary conventions: rhyme schemes, metre, and the like. If we know that an English poem rhymes, and we see "though" and "know" at the end of successive lines, we can imply they rhyme. We can see how Latin accented and deemphasized certain vowels and syllables based on how words fit the meter of a poem. There are also morphological conventions: things that just "sound right" for a language. No matter what scripts you write them in, aussprechen sounds German and "Masahiro Sakurai" sounds Japanese. As we develop translations, conflicts with accepted morphology can raise a red flag, letting us know we need to change something.

It depends. Often times, one will find that different historians spell the same name differently, and that is sometimes an attempt to more accurately reflect the historical pronunciation of a name. However, many names simply aren't pronounced by most people in the same way that they would have been pronounced in their time. /u/CommodoreCoCo 's answer covers that.

"Caesar" is probably the most famous example of a name that we pronounce differently from how it would have been said in the past, despite knowing how it would have actually been pronounced. In ancient Rome, Caesar would have been said with a hard C rather than a soft C, making it's pronunciation more similar to the German "Kaiser" than the English "See-zer." However, one rarely hears English speakers, even historians, use the hard C. There was a thread a while back on this.

Follow up question - If they are not the same, at what point, or where in the world, did they begin to become similar to our modern pronunciations?