House or an apartment needed for a couple.
Location inside the square of:
Bad Säckingen - Winterthur in the north.
And about Wollerau - Zug - Zofigen in the south.
With Zurich city area excluded.

Budget around max 2 kilo CHF including nebenkosten.
Small own yard preferred and dogs need to be allowed.
Needed asap.

You will be hard pressed to find anything in Wollerau/ZG at that price, let alone a place with a garden.

However, for some folks the low taxes in those areas more than compensates for high rents - but you would need to assess that given your own real numbers.

With dogs - especially in the plural, your need for a garden, and your price range it would be wise not to limit yourself to any specific area - look for an appropriate house and a landlord willing to accept you first, no matter where that turns out to be. Be prepared to commute longer than average.

Best of luck.