Travel from USA to India with $8K cash

There are two distinct aspects of this question

1. Can you carry $8,000 while leaving the US?

US CBP website says that

How much currency / money / monetary instruments can I bring into the U.S.?

There is no limit on the amount of money that can be taken out of or brought into the United States. However, if a person or persons traveling together and filing a joint declaration (CBP Form 6059-B) have $10,000 or more in currency or negotiable monetary instruments, they must fill out a "Report of International Transportation of Currency and Monetary Instruments" FinCEN 105 (former CF 4790).

Please be aware, if persons/family members traveling together have $10,000 or more, they cannot divide the currency between each other to avoid declaring the currency.

For example, if one person is carrying $5,000 and the other has $6,000, they have a total of $11, 000 in their possession and must report it on a FinCEN 105. If a person or family fails to declare their monetary instruments in amounts of over $10,000, their monetary instrument(s) may be subject to forfeiture and could result in civil and or criminal penalties.

Thus, you are entitled to carry $8,000 when travelling together without any paper-work.

2. Can you bring $8,000 into India?

According to FAQ on website of Reserve Bank of India

How much foreign exchange can be brought in while visiting India?

Ans. A person coming into India from abroad can bring with him foreign exchange without any limit. However, if the aggregate value of the foreign exchange in the form of currency notes, bank notes or travellers cheques brought in exceeds USD 10,000 or its equivalent and/or the value of foreign currency alone exceeds USD 5,000 or its equivalent, it should be declared to the Customs Authorities at the Airport in the Currency Declaration Form (CDF), on arrival in India.

Unlike US, India makes no mention about people travelling together and just states the limits as "per person". If both you and your spouse hold $4,000 each, you should be fine. Note that even if you were travelling alone, you could bring $8,000 into India but in that case you would have to declare it at the customs.

Assuming you are merely transiting in Doha, you don't have to worry about currency import/export regulations there. IATA states they have no limits on currency anyway.