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.