Q&A: Which top countries have their own alphabet but have transliteration problem in Latin characters?

I don’t know if Israel counts as a “top country” in this regard, but it does have a problem with transliteration of Hebrew into English (or in Roman characters in general).

Essentially, the problem is one of what is known in information theory as information loss: in the process of converting from Hebrew to English, information about the spelling of words (including the distinctions between letters that are essentially homophonic, such as ḥet and khaf, tet and tav, kaf and quf, samekh and sin, etc.) is lost, so if you were to try and convert back to Hebrew, you wouldn’t be able to reconstitute the original spelling—even if you were a native Hebrew speaker and recognised the words, as in the following example, from a poem of homophones titled Modern-day Ecclesiastes:

Is it possible to write Hebrew in Arabic script and vice versa?

Hebrew in Arabic—in principle, yes: Arabic has a direct equivalent of every Hebrew letter, plus six more. However, there are four caveats to this:

1. The Hebrew gimmel, by default, is a hard g sound—whereas in Arabic, the equivalent letter (ج – jim) is like a soft g (/dʒ/), like the English j. However, it could be decided that it is pronounced like a hard g, as in Egyptian Arabic.
2. The Hebrew vav sounds like /v/, but its equivalent Arabic letter, waw (و) is pronounced like the English w. However, it could be decided that it is pronounced /v/, since Hebrew has no /w/ sound.
3. The Arabic script has no hard /p/ letter—however, the Persian script (which is a near-identical derivative) does (پ)—so that could be used.
4. The Hebrew tzadi is pronounced /ts/, but its Arabic equivalent, ṣād (ص) sounds like /s/—so one would have to decide that it is pronounced /ts/.

How does the Ancient Phoenician Paleo Hebrew tongue differ from modern Hebrew? (It was only written in consonants)

As Aaron Christianson has pointed out, script and language are not the same thing: the Palaeo-Hebrew script is one thing, and the language was another—just as the Roman script is distinct from, say, the English or French or German language.

Q&A: How would paleo-Hebrew spell “Joshua”?

It would be spelled iod-heh-shin-ayin—which, in Paleo Hebrew (or the script of any Canaanite language of that period) would look somewhat like this (from right to left):

(in those days, the custom was not to use the letter vav as a quasi /o/ or /u/ vowel)

In Square Hebrew (i.e., the script adapted from the Aramaic in the Second Temple period and still in use today), those same letters look quite different, of course:

י ה ש ע

Q&A: How are modern Hebrew and Arabic related to ancient cuneiform?

The Hebrew and Arabic scripts are not at all related to ancient cuneiform.

Cuneiform writing began as a pictographic script and was in use by the various Mesopotamian kingdoms from around 3500 BC to around 900 BC. It was then gradually phased out during the Neo-Assyrian Empire (ca. 900–600BC) in favour of the 22-character alphabet that had been used by the Israelites, Phoenicians and other Canaanites since around 1500 BC (finally abandoned around 100 BC)—because that was clearly more efficient and easier to use.

How does the Ancient Phoenician/Paleo Hebrew tongue differ from modern Hebrew? (It was only written in consonants)

As Aaron Christianson has pointed out, script and language are not the same thing: the Palaeo-Hebrew script is one thing, and the language was another—just as the Roman script is distinct from, say, the English or French or German language.