Learn Arabic (Modern Standard): الحلاق في الصالون.

In Arabic, present tense sentences do not use a word for "is" or "are." This is called a nominal sentence. You just put the subject (الحلاق) next to the phrase describing it (في الصالون) and the "is" is naturally understood.

Why do we pronounce the "l" in the "al-" of the first word, but skip it in the last word?

This happens because of Sun and Moon letters. The letter Haa in الحلاق is a Moon letter, so we clearly pronounce the "l" sound. However, the letter Saad in الصالون is a Sun letter, so the "l" merges into the S sound, making it pronounced as "as-saloon" instead of "al-saloon."

What case ending does the word for salon take here?

Because it comes directly after the preposition في (in), the word الصالون is in the genitive case (majroor). If you are reading with full vowel endings, it would take a kasra (an "i" sound) under the final letter, making it "as-salooni."

The word for salon looks and sounds just like English. Is it a borrowed word?

Yes, الصالون is a loanword adopted from European languages. Arabic frequently adapts modern international words by spelling them out phonetically and simply adding the Arabic definite article ال (the) to the beginning.