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Your area is most equivalent to our USDA zone 9. The problem that you will run into will be that the pine tree will prevent most grasses from growing in the area.

Unless the tree is removed, completely, you will expect very little to grow around it.

This article will give you advice on the best grasses that can survive in your climate:

https://www.gardeningknowhow.com/garden-how-to/gardening-by-zone/zone-9-10-11/zone-9-lawn-grass.htm

According to their description, it will control weeds at seeding and allow the new grass to flourish. I would check the label, though, to make sure perennial ryegrass is not listed as one of the plants it controls. It does control nimblewill and bentgrass.

https://www.gardeningknowhow.com/plant-problems/weeds/nimblewill-control.htm

Typically, you want a good mixture of grasses to avoid bare spots throughout the year. This article will help you to find grasses for that area:

https://www.gardeningknowhow.com/garden-how-to/gardening-by-zone/zone-6/zone-6-grass-seed.htm

All lawn fertilizers (and all fertilizers in general) are made with the same chemicals in different ratios. Many of which are found in human and pet food. Though they are not safe to consume as they are, they will not harm your pets once applied in the correct amount. Your choice should be an all purpose lawn fertilizer, and that is about the most you need to worry about lawn fertilizers.

Here is an article that will help:

https://www.gardeningknowhow.com/lawn-care/lgen/lawn-fertilizer-tips.htm

This article will help you with dog urine issues:

https://www.gardeningknowhow.com/lawn-care/lgen/dog-urine-grass.htm

You'll need to kill off your existing grass first. This article explains natural ways of killing grass that, although not particularly labor-free, may be a good alternative to chemical methods: https://www.gardeningknowhow.com/plant-problems/weeds/how-to-kill-grass-naturally-kill-unwanted-grass-in-your-yard.htm

If you want to use a chemical, choose a nonselective herbicide with glyphosate in it.

Here's info about groundcovers for Zone 5: https://www.gardeningknowhow.com/garden-how-to/gardening-by-zone/zone-5/ground-covers-in-zone-5.htm

https://www.gardeningknowhow.com/lawn-care/lawn-substitutes/creeping-thyme-lawn/growing-creeping-thyme-lawns.htm

https://www.gardeningknowhow.com/lawn-care/lawn-substitutes/chamomile-lawn/growing-chamomile-lawns.htm

I would not do this with calcitic or hydrated lime. You can do so without ill effect with Dolomitic lime, though.

This article will help you with the care of a freshly seeded lawn:

https://www.gardeningknowhow.com/lawn-care/lgen/lawn-seeding-tips.htm

I would wait a couple days to make sure the product has had time to reach the roots.

https://www.gardeningknowhow.com/plant-problems/weeds/conventional-weed-killers.htm