Typically, the best type to trim bushes is when they’re dormant. Winter and late fall, potentially even early spring are the best times to trim your bushes. These seasons allow you better access to the branches, as all leaves are gone and the branches are more brittle, making the trimming process easier. 

Don’t pick up dirt after aerating. Even though it's tempting because soil cores are incredibly unsightly, these cores and the dirt they leave behind are ecologically important. This is what keeps your grass looking beautiful and growing healthily. The soil cores need two to three weeks to decompose, but you can swat them with a rake if their natural shape bothers you.

No, you cannot throw branches from your neighbor’s tree back into their yard, nor can you discard leaves or other plant matter that originated from their yard back over there if it happens to land in your yard. You are responsible for cleaning up plant matter that made its way into your yard.

If you want to take the decomposition process into your hands to speed it up, water and sunlight are key. Of course, you can’t control sunlight yourself, but you can rake leaves into piles in sunny spots in your yard. In terms of using water, you can also moisten the decomposing leaves with water, whether by pouring some from a watering can from time to time or using a hose to moisten them.