Is the call to prayer (adhan) obligatory for every prayer when praying alone due to the absence of congregants or a nearby mosque, and is it permissible to pray in a place that dogs enter?

First: The call to prayer (adhan) is not obligatory for someone praying alone, rather it is sunnah for him. If he raises his voice with it, it is better, otherwise there is no harm.

Second: The passing of dogs through a place does not prevent prayer there, because their mere passing does not render the place impure. The hadith of Ibn Umar (may Allah be pleased with them both) indicates this. The issue of dogs urinating in the mosque has been addressed by stating that it did not happen inside the mosque, or that this was before the ruling on the impurity of dogs, or that it was in the early days before the command to honor and purify mosques.