Yes and no. Drivers are not required to pay for a seat directly. Teams choose and hire their drivers themselves.
Some drivers are chosen based purely on talent and offered a salary by the team, others are chosen based on a mix of talent and cash/sponsorship money they bring to the team.
Many teams will opt for a less talented driver that brings money with them to help fund the team, where as some teams pay their drivers millions of dollars per year.
All drivers have to pay for a superlicense in order to be eligible to drive in Formula 1. This superlicense costs $12,800, plus $1,280 for each championship point earned the previous year., meaning some drivers have to pay over $1million for a superlicense while others only have to pay $12,800!
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All 24 cars enter 'Q1', each driver puts in a lap, the top 17 go through to 'Q2' (the drivers that didn't made Q2 have already earnt their starting position depending on who was the slowest (24th place) up to the quickest). In Q2, the fastest 10 enter Q3 (the drivers that didn't make it to Q3 have now also earnt their starting positions). At the end of Q3, whoever has put in the fastest lap is on pole position (they start 1st in the race the following day) and the other drivers positions depend on how fast they are.