1. Where is the restroom?
2. Where is a restroom?
[Can we use both expression? Do you use only #1? Does the restroom mean the specific restroom? Or is 'the restroom' in generic use? Or does the restroom mean the specific restroom near here, which people know?]
3. Where can I wash my hands?
[Can we use this sentence in another person's house or apartment? Can we also use this sentence when we are in a department store or at school?]
All three are acceptable. "The" restroom may imply an assumption that there is only one available for use. "A" restroom implies that there may be more than one, but you only need to use one at a time.
Sometimes we use the phrase "wash my hands" as a euphemism for the need to use the toilet, but, if you literally mean that your hands are dirty, any source of water and soap may suffice, such as the kitchen sink or an outdoor tap.