Well, the Azeri puppet state would have to be incorporated into the USSR eventually, you know, cause there's already an Azeri SSR. This would anger Iran even more, and I'd take longer for the two nations to have good relations again. USSR could start using Iraq as a substitute for Iran, and this leads to an interesting turn from our timeline. More specifically the Iranian Revolution, Iran-Iraq War and Gulf War. The US would have no one to side with, as the former (Iran) is fervently opposed to America in any form, while to latter is a Soviet ally. The war would probably end with a pyrrhic Iraqi victory, though Hussein wouldn't achieve much on the negotiating table. The US would also be more restricted in it's reaction to Iraq taking Kuwait. They'd probably try to solve the crisis through negotiation, and if there was conflict, US forces could take more casualties due to the Soviets being invested in Iraq heavily. Saddam would still be forced to leave Kuwait though, just with no repercussions. No matter if it'd be the USSR or Russian Federation that exists in 2003, Bush wouldn't dare to invade Iraq, therefore preventing it's destruction and subsequently rise of ISIS (in Iraq at least). ISIS might not exist in Syria either, as Iraq, being more friendly to Syria in this timeline due to a common ally, would intervene in the civil war. Unlike Iran, Iraq and Syria share a border so Iraq could directly deploy it's military and destroy the FSA much quicker.
In Soviet interior politics, the Nagorno-Karabakh conflict would still occur, but it's more likely the Kremlin would transfer the region to the Armenian SSR due to Azerbaijan being much larger than in our timeline. What's more important is that the USSR would be able to extend its influence much farther south, by having airbases in this territory. As another note, the Soviet war in Afghanistan could last longer, with Iraqi support, lasting to at least the mid-90s, depending how the USSR itself does.
As for the Kurdish Republic, it could either collapse quickly and be reincorporated, or be a puppet state for the USSR. If they are the latter, Iraqi Kurds could secede and join it during the 1968 coup. The Baathists would eventually have to recognise them to not risk war with the Soviets.