While it could be ADHD, it could also be a number of other things (including being a tempermentally-difficult but completely normal three-year-old boy). ADHD is characterized by extreme impulsivity, inattentiveness, and certain social deficits. It's caused, in part, by inefficient dopamine receptors, which probably explains the strong correlations between ADHD and later substance abuse. Since the behavioral features of ADHD are also similar to features of normal (but immature) toddler behavior, it's hard to determine conclusively without a great deal of observation. Formal diagnosis often doesn't happen until later (school age) for this very reason, when some cognitive maturation should have occurred (and reduced these behaviors). Often ADHD (like learning disabilities) is genetic/occurs in within families. Finally, there are many other disorders that cause behavioral traits that resemble ADHD (or occur alongside it), so there may be other things going on.
(+ normalish toddler behavior)
(# atypical/possible red-flag)
+ does not self entertain
+ aggressive at times
# does not listen (can be for a variety of reasons, but if it's pervasive, that's worth looking into since most kids WANT to please adults at least some of the time)
+ throws tantrums
# makes extreme messes (if deliberate/vindictive)
# lies (if it happens frequently that's suspicious)
# extremely needy for adult interaction (often a feature in attachment disorders)
# can't listen
# breaks things (if vindictive as opposed to accidental)
+snatches toys out of hands and runs
+makes a mess
# takes food off of my son's plate and eats it (this one's pretty odd--does he do this a lot?)
+begs, whines and cries for everything (his dad is reinforcing it, so it's going to continue)
Worth noting, regardless of whether or not he "has anything", he's probably not getting the kind of parental attention he requires (His dad is also overworked and probably very depressed.
His mother is busy cooking and cleaning.) If he doesn't have a solid emotional connection with his primary caregivers, that can lead to a lot of insecurity/acting out.
The important thing to ask, is are his parents concerned about these behaviors? What do they think? If this behavior is occurring in ALL settings and they're concerned about it, they may wish to have him evaluated to rule out a problem, but ultimately the choice is theirs.
Regardless of what THEY are willing to do about his behavior, you'll have to decide what YOU want to do--if you want to keep watching this kid, you may want to explore some ways to proactively deal with some of his more difficult behaviors (whether he has ADHD or not), since a lot of the interventions that are commonly used with kids who DO have ADHD are helpful in working with any kid who seems to be having difficulty managing their own behavior. Good luck!