Iran as a country cannot be ignored or underestimated..especially by Pakistan.
Literally, Pakistan has an oil/gas giant sitting at its doorstep that can help alleviate most of Pakistan's energy problems esp in case of war. Having a neutral Iran who is willing to supply Pakistan regardless of posture viz a viz india is a trump card against any economic blockade India can envision.
Next is the sectarian card. Iran and Saudi are rivals in the Islamic world and Pakistan given the sectarian problems its facing can become a playground for these religious forces. Esp when one is at your doorstep and the other being an all weather ally. With the evergrowing importance of the CPEC (next door to Iran), you want as little to stir up the pot as possible.
Regarding your last two points, Pakistan can take those up with Iran and work towards resolving such issues wrt. spies, but cannot dictate terms or go on the aggressive against Iran. Pakistan has a weak hand geo-politically (at the moment) and should tread carefully with its relations in the region.
You may think differently, but this is my unbiased opinion of the current situation.
Easier said than done...
A very simplistic view of the situation.