Shade and dogs and heavy foot traffic will make growing grass a bit more difficult. Hopefully your trenching has made a good improvement removing the standing water. A large bag of grass seed is about 50 bucks. Buy one. Scatter it evenly right on top of the ground as is. Look for a shade tolerant blend if you think that is best. It will probably be a fescue blend for your area from the looks of things. Now is a good time to sow grass seed up north.

Does your yard have an overall pitch to it? Or, is it so flat that you can't tell? If the former, you probably need some precision grading that makes sure no water collects during a rain or stands anywhere after one. If the yard is so flat you can't tell if there's pitch, it needs to be measured with instruments or have someone with a keen eye and knowledge of drainage look at it.

Nothing to do with your question, but the tree in the first photo is in desperate need of having some lower limbs removed. It's too close to the house to be having limbs that low becoming permanent parts of the canopy. Lower limbs that don't need to be there only contribute to the excessive shade and inability to grow plants -- like grass or groundcover -- below the tree.