Is convincing clients of competing companies to switch to us by offering lower prices and better quality considered like selling over a brother's sale, and are gifts given to companies or individuals within them without their company's knowledge considered bribery?

As long as you know that these employees are agents for another competing company and have a contract with it, then deliberately convincing them to leave their company and contract with yours falls under the meaning of "selling over one's brother's sale," because ijarah (leasing/hiring) is the sale of a benefit. This includes the case where the contract has ended or there is mutual agreement to renew it. Sheikh al-Islam Ibn Taymiyyah holds that hiring over one's brother's hiring is more harmful than selling over one's brother's sale. He compares a person bidding over his brother's bid to proposing over his brother's proposal, and he differentiates between situations where there is firm inclination and those where there isn't. This prohibition extends to renting, borrowing, and seeking employment over one's brother's request.