3 important tests would be calcium hardness, pH and Total Alkalinity. Make adjustments as needed. I would not worry about a stabilizer.

Chems should be dissolved in a bucket of pool water individually and added to the pool a little at a time. Muriatic acid is best to control pH and Total Alkalinity. It must be diluted and not poured in straight as it can harm the finish.

I generally hold off as long as possible before adding chorine and, I will only use liquid chlorine initially. Most likely your area does not have a high demand for chlorine as we are in the winter months. So, I would say you're ok.

What type/form of chlorine was added?

Brush at least twice a day for 2 weeks. Brushing even more frequently is perfectly fine.

The dust you see is mostly the plaster dust when you brush. Keep an eye on your filter and clean it when pressure builds and the flow slows down. The brushing will cause the dust to be suspended and the filter will pick it which is its job.

No auto vacs for 2 weeks. Manual vacuuming should be done with a vinyl vac head that has brushes to protect the finish. No salt for at least 30 days. Go longer if you can.

The plaster will take 28 days to fully cure but the 48-72 hours after filling and start up are critical to balancing.

Thank you so much!