Is the mispronunciation of foreign words especially likely in English?

The problem is that there are a number of hidden assumptions behind this question that need to be picked away before the question can even be posed. Let me take them one by one.

Are there other languages out there, more phonetic than English,

This apparently refers to the English writing system, which is notorious for misrepresenting English pronunciation, as well as the pronunciation of any other language. English spelling is often said to be "not phonetic", though that's popular and not technical usage; phonetic science is actually about speech sounds only, not about writing.

in which the sound of foreign words can be specified adequately?

The writing systems of every language, like their phonemic systems, are unique to and adapted to their own language. All of them are poor at representing sounds and sound combinations that don't occur in their own language; that's not what they're for, after all. What can't be pronounced can not be represented accurately in writing.

So you shouldn't look to other languages; you should look to phonetic science. The International Phonetic Alphabet is exactly what its name suggests -- a standard alphabet to be used to represent all human speech sounds, at the alphabetic (basically the phonemic) level.

For example, is it the case that when Arabs move to America their names are frequently mispronounced by people reading the English spelling; while Arabs who move to Japan are able to write their names in Japanese such that Japanese people frequently pronounce them well?

No, that's not true. When Arabic speakers with Arabic names move to America, it's true that most Americans can't pronounce their names -- no matter how they're spelled -- not because of spelling, but because they don't know how to say many of the sounds. Arabic is full of sounds that don't occur at all in English.

When people who are not Japanese and don't have Japanese names move to Japan, Japanese people often can't pronounce their names, either, because they don't know how to say many of the words. Japanese has an extremely restrictive phonology -- virtually all syllables are Consonant + Vowel, for instance, with no real clusters and a small phoneme inventory -- and therefore Japanese speakers find many English words difficult if not impossible to pronounce.

(Squirrel is the English word my Japanese students found hardest when I was teaching ESL.)

There is a special syllabary in Japanese writing that is reserved for foreign names; but it does not represent their pronunciations -- it just indicates that they are foreign.

(My Japanese students had a lot of trouble with my last name, too -- the kana transcription of Lawler /lɔlər/ comes out as Roreru.)

Is English one of the hardest languages in which to specify the sound of a foreign word?

No. As you see, this question isn't about the English language, but its writing system. The writing system is independent of the language; it's just one method of representing the spoken language.

The spoken language, the evolved language, the living language, the one that everybody learns before they go to school, whether they ever go or not, is the real language, and it would be the same language if it were written differently.

I will say that the English writing system is one of the worst-adapted in the world; however, much the same can be said of the Japanese system. And there are other things besides phonemes that writing systems need to represent.

Whether people can pronounce a foreign word depends more on if the sounds are familiar than on if they have a familiar way to write them. Many Japanese speakers are well aware of the difference between the letters "l" and "r"; that doesn't make it easy for them to hear the difference between English "l" and "r". Similarly, it takes a few minutes to teach an English speaker that French has one vowel written "u" and another written "ou". Teaching someone how to distinguish the sounds of these vowels takes a lot more work.

No two languages have exactly the same sound systems: this is why people often have noticeable "foreign accents" when they speak a non-native language.

The differences between different languages' sound systems is a major factor that contributes to "mispronunciation" of foreign names. This factor will be present no matter what writing system is used.

Writing systems can also contribute to uncertainty about how to pronounce a name, but

1. English isn't unique in having an ambiguous and underspecified writing system. It's towards the "complicated" end of the spectrum, but in most languages there are some kinds of spelling-to-sound ambiguities, particularly in areas like stress, tone or vowel sounds. Italian has a very straightforward writing system compared to English, but it often doesn't mark contrastive stress, so there are two Italian words written "principi" that are pronounced differently (one is stressed on the first syllable, the other on the second syllable). In my experience, any language you can find that is said to have a "phonetic" writing system will actually have some complications like this.

2. In a number of languages (not all), it doesn't even matter if the usual rules for spelling give a strong indication of the pronunciation of a word, because personal names can be exceptions to the usual rules. E.g. a number of people in Spain have the name "Jessica", which might not be pronounced the "regular" way you'd expect from the spelling, /xeˈsika/, but as /ˈʝesika/ instead. A relevant Spanish SE post: Pronunciation of Boca Juniors

3. Cultures where it is common to alter the spelling of foreign names to conform to the native writing system are probably also likely to alter the pronunciation of foreign names to conform to the native sound system (which you might consider "mispronunciation"). An example: in Lithuanian it is traditional to re-spell foreign names according to Lithuanian spelling conventions, but this process also adapts the sounds to the closest Lithuanian sounds. It doesn't preserve all the distinctions that the foreign language makes. According to the following document "Lithuanisation of Personal Names of the Polish Minority in Lithuania", by Justyna B. Walkowiak:

   Polish letters with diacritics that do not exist in Lithuanian (ć, ł, ń, ś, ź) are replaced with the same letters without diacritics (c, l, n, s, z respectively, e.g. Ćwikliński – Cviklinski(s), Paweł – Pavel(as), Jasiński – Jasinski(s), Śniadecki – Sniadecki(s), Kuźma – Kuzma). Since most of these diacritics (except for ł) mark palatalisation in Polish, such spelling does not reflect standard Polish pronunciation.

So pronouncing names from a foreign language "correctly" (the way a native speaker of the foreign language would) is difficult with or without respelling.