Using Krosan Grip

[TrialByFire]

That last answer just brought to mind another k-grip non-related question about priorities.

If I declare attackers and the opponent wants to animate his Mishra's Factory to block it, do I get priority in between the declare attacks step and the declare blockers step to swords to plowshares his factory before it gets a chance to block my guys?

Yes he must activate in the end of the declare attackers step to be able to block. You can't go to the declare blockers step until both players pass priority in succession, so I assume you would attempt to StP the factory when you are given priority at this time.

[Anusien]

If I declare attackers and the opponent wants to animate his Mishra's Factory to block it, do I get priority in between the declare attacks step and the declare blockers step to swords to plowshares his factory before it gets a chance to block my guys?

Yes. You can't get out of a step without both players getting a chance to play spells or effects.

[mugs]

Question about Relic of Progenitus, and K-gripping it?

Can I respond to opponents tapping or mana activation and stop either effect?

not sure how this one works. I thought it was simple enough but opponent has argued otherwise - that they have priority and I'm basically SOL.

[quicksilver]

You can't respond to mana abilities ever.

And if you destroy something in reponse to them using an ability from it the abilitiy they played still happens.

[Elficidium]

Question about Relic of Progenitus, and K-gripping it?

Can I respond to opponents tapping or mana activation and stop either effect?

not sure how this one works. I thought it was simple enough but opponent has argued otherwise - that they have priority and I'm basically SOL.

Removing the source of an ability does not stop the ability.
Next time you have priority you will be able to grip it.
And that would be the ideal time, if he doesn't activate it right away and wants to play something else/go to the next step you can grip it. Grip it hard.