I want to explore, but I have to constantly mange relationships with the factions so that they wont get too negative.
[...]Will constantly paying bribes (so far i only payed only 1 for 100k) would be just as or more effective in terms of generated money as shutting down the production industries
Ever heard of AI core bribes? You go to, say, a Hegemony colony with a military base, speak to their commander, then you arrange a deal.
I don't really know (I didn't see anyone mentioning it) how to stop the expeditions or how to mange them.
If you keep your market share low, expeditions will not happen or be infrequent. This typically means sticking to just farming and maybe mining for sources of revenue. After that, you just have to get used to taking the expeditions on.
Additionally, Luddic Church and Hegemony will send expeditions your way for having free port switched on, while Hegemony will also send AI inspections for using AI cores in industries.
Is it possible to shut down the production industries on planets (size 7 and size 6) so that they would still generate some money (just to cover officer and crew salaries) but wont trigger the expeditions?
Yeah, just dismantle all the money making industries. Should be enough.
Will Hegemony send AI expeditions regardless of me using AI cores on the colony or just having them in the colony stash (personal storage)?
Keeping them in storage is safe. Using just a single one, though, puts your colonies at the risk of inspections.
Do you know more efficient ways of solving these issues (managing faction relationships while having colonies and having time to explore the sector)?
It's best to colonise multiple planets in the same system, so that you can stack patrol buildings on them and have the entire system (and all the colonies inside) protected fairly well. You can try bribing the factions initially, but later on its just better to farm AI cores.