all items can be found on walks, no trick, just luck.
The siren is only a little easter egg (easter egg as in little joke) it means nothing to the firemen hat, accesories have no influence on items, trust me, I hacked the game and there is no script for anything increasing item rate. but there is ONE trick to get better items, and that is going to where ? boxes WOULD be, you'll either find trash, bad food or a present, the present gives a better item then the ? boxes
HOPE IT HELPS.
Jack russel is obtainable via jack russel book (an etc item found on walks)
dalmation is obtainable via fireman hat (an accesory found on walks)
All pets and backgrounds are obtainable on every version.
OH yeah, another tid bit of info
I used to be cool77, used to post in the early days (2006) of nintendog Q&A where people was crazy and didn't talk about nintendogs, if they did they talked if dogs can grow or breed (which IS NOT POSSIBLE by the way), I left after this Q&A got some sense restored, anyway, that's enough storytime folks! XD

If you have any questions say here and I'll try to respond as soon as possible.