About the part where they say that will charge your payment method, when you terminate your relationship with the company, you should send registered mail withdrawing authorization to charge your credit card thereafter, and you should notify the credit card company that further charges are not authorized. This won't change your legal liability, it just changes whether they have to first prove that you are legally liable before collecting the money.
It is problematic that only the company has the right to terminate the contract, but you can treat that as a harmless omission of wording on their part. Surely they did not intend that you are perpetually bound to the terms of the contract and the obligations in section 11. Of course, you have to stop using their services in order to terminate the contract. Given the nature of their service, you can send a notice (registered mail again) to them informing them that you are terminating the contract and will not be using their services.
If then some time in the future you do hire a pet walker that you met through the service, they might come after you for the $1,000 (although, at their sole discretion, they can increase that amount to $20,000 or anything else, see section 2). If you have prevented them from charging your credit card, they would have to get a judgement against you, meaning they would have to go for arbitration. Or, if they did charge your credit card, then you would have to initiate the proceedings to get your money back. That is, unless you opted out of arbitration within 30 days of starting the relationship. Let's assume you did opt out.
The question would then be whether a perennial "no-compete" clause would be enforceable against you. Employees are often bound to no-compete agreements prohibiting them from doing the same work in the same area for a certain period of time. A lifelong no-compete agreement would be unenforceable, at least in court, but a 6-month no-compete clause could easily be. In Medix Staffing Solutions, Inc. v. Dumrauf, the court recites standard legal requirements of enforceable no-compete agreements, especially that they be "reasonable and necessary to protect a legitimate business interest of the employer". In such a case, the court would consider "the hardship caused to the employee, the effect upon the general public, and the scope of the restrictions", and the company would have to show that the "full extent of the restraint is necessary for protecting its interests". It is in principle possible that the court would find such a clause unenforceable (owing to it imposing a life-long obligation on you). This could be a new area of law: customer non-compete clauses.
One other point about that clause is that it is just a penalty clause, which is unenforceable (calling it a "referral fee" doesn't change the fact that it is a penalty clause). It can't reasonably be interpreted as a liquidated damages clause (which is enforceable). $100 might be.
You might rely on the mercy of the court, hoping that they would find such a perma-ban clause to be "unconscionable" and unenforceable; in principle, the arbitrator could find the same thing. An alternative would be to accept the conditions in the contract, and never hire that dog-walker on your own.