HCAD Appraised Property Value Protest Question for New Owner

I recently bought a home last year in Harris County. The previous appraised value (2023) was $300k. I bought the house for $420k in July 2023. This week, I got the 2024 appraised property tax value, and it was $400k. I never submitted / notified the county of my purchase price. Since July 2023, house values have decreased in my area as a result of interest rate increases and a weakening real estate market.

Three questions:

1. Is it worth it to protest the value if comps for similar houses in my area is closer to $340k than $400k? Or should I be happy that the appraised value isn’t $420k? (I assume in 2025, since I was granted homestead almost right away when I bought the house, the $400k will be subject to the 10% homestead cap?)

2. Does HCAD know how much I paid my house for, or was it an calculated guess? I know Texas is a non disclosure state but I am certain there are ways to find this info — I’m not sure about the specifics as to how.

3. (MOST IMPORTANTLY). Is there a risk of my protest increasing my apprised value, say to my purchase price?

I am happy to protest the amount myself before HCAD appraisal board myself and do the research on comps and supporting documentation of deficiencies of the property, but I wonder if it’s a moot point because if HCAD would ever ask for my closing statement / purchase price, it’ll indicate the $420k.