Learn Arabic (Modern Standard): أين رخصة السائق؟

This is the main rule of the Idafa (possessive) structure! The first word (the thing being possessed) never takes ال (the). It automatically acts as if it has 'the' because it is connected to a specific owner. So we say رخصة السائق (literally: license of the driver), not الرخصة السائق.

How should I pronounce the end of رخصة here?

You must pronounce the 't' sound. The word رخصة ends with a taa marbuta (ة). Normally, if you stop on this word, it sounds like an 'a' or 'ah'. However, when it is the first part of an Idafa, the rule is that you must pronounce the 't' to smoothly connect it to the next word. It sounds like 'rukhsat as-sa'iq'.

Does the word السائق (the driver) change its ending because it is the owner in this sentence?

Yes, if you are reading with full formal vowels. In an Idafa, the second noun (the owner) is always in the genitive case. For most singular words like السائق, this means adding a kasra (a short 'i' sound) to the last letter. However, in normal conversation, native speakers usually drop this final vowel sound at the end of the sentence.

Where is the word for 'is' in this sentence?

Arabic does not use the verb 'to be' (is/are/am) in the present tense. You simply put the question word أين (where) next to the phrase رخصة السائق (the driver's license), and the 'is' is naturally understood.