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Can anyone recommend which car hire company I should use from Lisbon airport to drive down to Algarve?
When I was travelling from the UK to Portugal ans needed to hire a car I used Zest Car Rental https://www.zestcarrental.com They are an agency which uses a variety of actual rental firms at each airport. I never had any problems with any cars I hired through them, and never had any issues with extra fees/charges after the hire.
As to the actual care hire firm, if I remember correctly, I have used both RentAuto and Centauro, which are still listed as options for Lisbon airport.
We have used Zest for a number of years and can recommend them without reservation. As said above they are a broker. Charges are transparent, customer service excellent. We regularly 'test the market' but always revert to Zest: especially competitive if there are additional drivers for which there are, in our experience, no additional charges.