Your house is utterly charming! As for the weeping cherry, I have many and they all have random dead branches which I prune off. Keep up with removing suckers.

You also might try posting this over in the Garden or Shrubs or Trees forums as well...you're probably not going to get the optimal answers in Lawn Care!

Any ornamental cherry tree is prone to a huge number of disease ad pest issues and for some reason, the weepers seem to be more susceptible to problems than the upright forms. Bacterial canker and brown rot are two of the most common issues. Generally not a long term plant in the garden for this reason :-(

Just prune off the dead limbs. That's really all you can do at this point. In winter, you might look into dormant sprays but I'd figure out what you might want to replace it with. Once they start going down hill, it's pretty much a non-stop ride. Sorry

Nice lawn!