Learn Farsi: دکتر معاینه‌ام کرد و یک داروی جدید تجویز کرد. - The doctor examined me and prescribed a new medicine.

In Farsi, it is very common to use attached (enclitic) pronouns for the direct object instead of the formal independent word مرا (ma-râ). With compound verbs like معاینه کردن (mo'âyene kardan - to examine), the pronoun attaches to the end of the non-verb part. So, معاینه + ام + کرد becomes معاینه‌ام کرد (mo'âyene-am kard - examined me).

Could I put را (râ) after the first direct object, 'me'?

Yes, but you would have to change the structure. If you wanted to use را, you would use the independent pronoun من (man - I/me) + را to make مرا (ma-râ). The sentence would become: دکتر مرا معاینه کرد (doktor ma-râ mo'âyene kard). However, using the attached pronoun -ام as shown in this sentence is much more natural in everyday speech.

Why is there an extra ی on داروی (dâru-ye) if the word for medicine is just دارو (dâru)?

This is the ezâfe (the grammatical link connecting a noun to its adjective). Because دارو ends in a long vowel sound (u), the ezâfe takes the form of a 'ye' sound to smoothly bridge the gap to the adjective جدید (jadid - new). It is written here with the letter ی.

Since 'a new medicine' is the direct object of 'prescribed', why is there no را (râ) after it?