I want to make sure I understand these cards correctly.

Does Smashing Ground work if all or some of the opponent's monsters are face down?

Can you equip Megamorph to your opponent's monster (it stays in my S/T zone), so that while MY Life Points are higher I can halve the original ATK of my opponent's monster?

Can Michizure be activated any time my monster goes to the Graveyard? Destroyed by battle? Card effect? Tribute? Cost?

Smashing ground will not work on your opponent's face down monsters. Only the face up monsters will be looked at to be destroyed.

You absolutely may equip Megamorph to your opponent's monster to halve its attack.

Michizure will only work if the last thing to happen is the monster sent to the graveyard. In addition, it cannot be activated in the damage step, so being destroyed by battle will not help. Costs are never the last thing to happen, so paying costs will not enable you to activate this card. This includes tributing for a tribute summon. Basically, you activate this card after your opponents uses smashing ground or sakuretsu to kill your monsters.

I am not for sre on this card. Hopefully someone else will awnser it.

D.D.

Thanks. You know, I was reading a post earlier (an older one I think) and someone commented about the chains in a link and the precision of actions that take place. He was right when he said it's important to understand the technical details of stages and steps and links of effects. (Hey, that rhymes.)

Smashing Ground as answered only searches for face-up monsters, if all monsters on your opponents side of the field are face-down you cannot even activate it.

You may equip "Megamorph" to your opponent's monster, but 'your Life Points' refers to the controller of "Megamorph". So if you have more Life Points than your opponent, your opponent's monster equipped with "Megamorph" has its ATK halved.

Michizure as said earlier cannot be activated in the damage step nor in between chains.