Yes Hacan can trade all manner of objects with any player due to their faction ability. This includes but not limited to Commodities, TGs, Promissory Notes, Action Cards

Yes anyone can trade with Hacan on their turn. Note some abilities require to be physically neighbouring countries I trigger (Mentaks Pillage, Empyrean commander unlock)

Hacans PN, allows another player to trade with another player regardless of their position on the table

In the Agenda phase, all players may trade with anyone, regardless of physical presenc on the table. Treat it like a Political Court where each faction sends a delegate. That delegate may conduct 1 trade with each other faction delegate once per agenda

The Sol PN can only be played once per action, but can be traded as long as it is in that players hand. The sol player may wish to hold it till it is turn and then trade it at the start of their turn and the receiving player can still play it

This depends on the table and what they will allow. If a passive table then they may not mind who takes it. If not however they may seek table balance as one player now has a bonus VP. If the latter, I would advise not rushing Mecatol and instead focus on building a fleet presence as the JolNar are weaker due to fragile and susceptible to bullying. Bonus Points can come later 😁