Hey Sam, I'm not a tax expert, but I am an agent and an investment property owner in Austin and surrounding areas. As far as your valuation, Texas is a non-disclosure state, so in theory the county tax assessor shouldn't have the information about what you paid... but in my experience they get pretty close. If they assess you for an amount higher than you paid, then you can produce your closing HUD statement and they will drop your valuation to what you actually paid. They will send you a letter asking you what you paid, but you can ignore that.
Since it is not your primary residence, they can raise your valuation by any amount they see fit. The 10% YoY cap applies to primary residences only (homestead exemption). You can protest your valuation or hire a company like Five Stone to protest on your behalf. I recommend you do that every single year.
If you're working with an agent here, they can help you determine the tax rates - they vary block by block. I've seen anything from 1.6% in unincorporated Travis County to nearly 3% in areas with special utility assessments, etc. Good luck!