Well, the ‘puzzle’ makes no sense. Who on earth would ask for “H2O” at a restaurant? Not even a die-hard Chemistry fan. And what restaurant would ever serve Hydrogen peroxide to its customers?

The answer given by ‘silent47’ is surely logical, but…

Consider the scenario where the defendant didn’t kill his wife, but, as a matter of fact, is very much aware that she’s actually dead (might have been killed by somebody else). The lawyer, in a last-ditch effort to save his / her client,  attempts to play the ‘doubt factor’ and makes the statement presented in the question.  In this scenario, the defendant knows for certain that his wife isn’t coming, and hence won’t look at the door for the ‘dead woman walking’. In this scenario, the jury’s  ‘guilty’ verdict would amount to gross miscarriage of justice.

Now, consider another angle: The defendant is actually guilty. The lawyer, aware that any statement (s)he makes in the Court is binding on the defendant, wouldn’t make any statement / claim without having consulted him beforehand. Had the lawyer discussed the ploy (of making the jury doubtful) with the defendant, (s)he would have trained the defendant to play his part in the Court room ‘drama’. (Otherwise, the whole ploy is useless.) In such a scenario, the defendant would have looked at the door as well and the sequence of events presented in the question couldn’t have arisen.

The ‘expected‘ answer seems to be 20, as suggested by ‘silent47’, but that answer requires the assumption that I’m coming home along the same road as I went to the Store, whereas it need not necessarily  be so. 🙂

The killer starts by entering the room to the north of his room and killing the person there. Then he comes back to the starting room (This is the only room to which he can come back, because there can’t be any dead body there as it was originally occupied by himself.) Then he proceeds to other rooms in the sequence  2, 3, 4, 8, 7, 6, 10, 9, 13, 14, 15, 11, 12 and 16, exiting from there. (See the figure for room numbers and path taken.) 

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Added later: He can take an alternate path with first move eastwards (to Room No. 2) and coming back and taking a path mirroring the first solution, along 1 -> 2 -> 1 -> 5 ->  9 -> 13 -> 14 -> 10 -> 6 -> 7 -> 3 -> 4 -> 8 -> 12 -> 11 -> 15 -> 16. 

See image for room numbers and the ‘psycho-paths’.

Assuming that “this property” in the question refers to the ‘squarishness’ of age only (and not the number of years since last squarish age being also squarish), we have to wait 14 years, when he will be 56 (7 * 8). His current age is 42 (6 * 7), and the last time his age was squarish was when he was 30. (5 * 6), the number of years passed since then being 42 – 30= 12 which is again squarish (3 * 4). 

The next time BOTH these conditions (i.e., age being squarish AND the number of years since last squarish age also being squarish) becomes true will be when he’s 110 yrs (if he lives till that age). 110 = 10 * 11, and the previous squarish age would be 90 (9 * 10), and the difference, 20 yrs (which also is squarish, being 4 *5)