Divine metamagic is a free action and the spell shaper ability does not have a listed action type, but increases casting time. You may perform one or more free actions while performing a normal action, so they will both work short of DM intervention.
Both abilities allow you to cast spells of a level higher than you can currently cast. Spell shaper does still require you to have the necessary casting stat, but note DMM does not increase caster level. So for instance if you added extend with spell shaper to a level 3 spell and applied DMM persistent, you would need a casting stat of 14 ( 10 + spell level + 1 for extend).
For the Orb of mental renewal or the Rod of physcial restoration (works the same for physical ability scores, same book), you would not need to make the caster level check. Character's casting a spell require the caster level check, activating an item that does not cast a spell and is not spell completion would not.
With all that said, I am not sure you really need DMM if spell shaper is available. One you said you are working towards DMM so I persume you do not already have it. Even with everything in place, you still have 2 more levels to level 12 for your next feat. Assuming you have extend and persistent spell, you already maybe able to move directly into spell shaper. Also at that point you will gain access to the heal spell which will remove all ability score damage. Your caster level check will be 14 at that point + you should at least have a bead of karma from the DMG for a +4 caster level for 10 minutes a day. So, activate bead of karma, cast all your buffs putting damage from each spell on a different ability score (not Wis though), and then cast Heal with a caster level of 18. You only have to worry about a 1 on the roll and only for one level. Get an orange Ioun stone or a spell that boosts you caster level and that is not a problem. Buy a second bead of karma to repeat the process if you fear dispel or disjunction. If you go epic, using your turning for the Spell Siphoning feat from the Defenders of the Faith web enhancement.