Learn Russian: Мы долго договаривались с клиентом. - We negotiated with the client for a long time.

Yes! The perfective form договориться means 'to reach an agreement' or 'to make an appointment'. But here we use the imperfective form договаривались, which focuses on the process of trying to reach an agreement. In English, we usually call this process 'negotiating'.

Why is there a reflexive ending -сь on договаривались? Aren't we doing this with someone else, not to ourselves?

The ending -сь (or -ся) doesn't always mean you are doing something to yourself. Here, it acts as a reciprocal ending, meaning a mutual action. When you negotiate or agree, two or more people are doing it together.

Could I use the perfective form договорились in this sentence?

Not with the word долго (for a long time). The word долго tells us that the action was an ongoing process, so we must use the imperfective verb договаривались. If you wanted to say 'We finally agreed', you would drop долго and use the perfective договорились.

Why is the word for client spelled клиентом here?

Because of the preposition с (with). When с means 'together with', it requires the noun to be in the Instrumental case. For a masculine noun ending in a consonant like клиент, we add -ом to get клиентом.