Appartment short term Osaka

Hi !
We are two people, with 2 dogs (1 shiba, 1 Shetland – 8kgs).
One of us speaking Japanese N3 level, and me N5. Both fluent in English.

We would like to find an apartment in Osaka or around. Even outside Osaka, as long as accessible to the center (I will be studying at ISI).
We were thinking to rent a short term apartment before finding a place for a full year at least.

But we have two troubles : we have difficulty understanding how to manage finding a place that accept both strangers and pets, if possible at an affordable place. And also it is difficult to know the cities / suburbs, where dogs could be happy (good walks at least, big park, a garden ?).

We are currently in Japan and we know some places are better to be with dogs than others !

Our questions : do you have websites, agency names to find apartments (short term) ? Is doing a short rental a good first move before finding "the one" ?

And if you have names of suburbs, or cities nearby "dog friendly" we take names !!

Thank you very much

by Zero_Border