No living thing should lie by the side of the road after an unfortunate accident. If you feel compassion for its remains, but do not have the resources -bags, boxes, gloves, plot to bury, etc- to take care of the body appropiately yourself, call this number and they will transport it to a crematorium for free. It is a public service.
For your own dog, that is your own responsibility and your choice.
You may have the body removed by the city but then it will dopose of it in a common grave. I am not sure if it is free or rather inexpensive. Lemme search…
EDIT: they have a 1999 line in Taipei City for general services, but I am not sure this one is included.
You have several options in terms of private burials or cremations. You can have the remins cremated and then keep it at home or in a columbarium or buried in a private plot. That can go from 5000 to 25000 to XX000…