Does seemingly random output imply irreversibility in hash function?
Now my question is if a hash function has this property: for small changes in inputs it outputs such numbers that will seem random so that from output it will be hard to determine changes in input, does it imply any of the above property specially property 1, i.e. irreversibility of cryptographic hash function? 


No. Randomness is hard to test and it's easy to use any of the non-secure random number generators to generate random looking output. So anything that is "non-random" is certainly not irreversible because of just that.


  Putting slight differently is it possible to train a hash function from input and output if outputs are not seemingly random so that for given output I will know input?


That's kind of the opposite. But if you just take $H'(x) = H(x) | 1$ then it should be clear that $H'(x)$ is as irreversible as $H(x)$, while the output is clearly not random.