I absolutely agree you should remove that tree. It's way too close to the house. When you do go to replace it be sure to research really thoroughly as to how big the new tree will get before you plant it. You need to plant it at least its eventual full height away from your house both to prevent damage from limbs falling and to keep the roots from invading your foundation and plumbing.
Two more thoughts - in your first picture, look beyond your tree to the tree in the yard across the street. Whoever pruned that tree knows what they were doing. It's really nice (although I have no clue what kind it is). That yard is a bit too "groomed" looking for me with all the shrubs trimmed into ball shapes, but that person either has good help or knows what they're doing. Maybe get to know them, and ask about a good tree to plant, and later, how to prune it or have it pruned. An example of really bad pruning (or none at all?) is the tree in the next-door yard with the little brick thingys around it.
Second, the beginning of summer is not the time to think about pruning trees, just for future reference. Sap is flowing full-on and some trees can literally bleed to death from major pruning at this time of year. Fall, winter or early spring before the leaves start to bud (dormant times) are all better times. So if you do leave the "maybe cherry" tree and try to recuperate it, wait until it starts to lose its leaves in the fall, at least!
