Is the homeowner responsible for the tree growing on the patch of grass near the sidewalk opposite to his/her house, just as he/she is responsible for cutting the grass on that patch? Any legal consequences?

I think it depends on your town/city laws. Where I am, I am responsible for everything to the street, including the condition of the sidewalk and the curb. And yes, there are legal consequences if not keep up and in safe condition. My city does issues tickets and fines.

Most cities require you to cut the grass on the parkway. You are welcome to feed the tree, trim it but you you are not legally responsible for it. If it dies the city pays to remove it. If it drops a branch on someone's head the city gets sued first, it's their property.