Don't hire the agent's inspector. Likewise, make sure you have your own agent, don't use the seller's. And if you're in court someday, don't expect one lawyer to represent both parties. Simple stuff.
It's similar to buying a car: don't ask the dealer's maintenance department whether it's a good car. Drive it to your independent mechanic on the test drive and ask them to take an unbiased look at it.
If you don't know a reputable home inspector, you could ask your painter, contractor, or handyman if you have a good relationship with someone who knows houses to take a look.
They are hired by the agent
Another complication to be aware of: Inspectors can face "blackballing" by banks or realtors if they scuttle too many deals by finding bad things. If you find an independent inspector, don't balk at paying a little more -- they may need it to make up for business lost from being honest.
Not exactly. Real estate agents do care about repeat business and referrals.
My home inspector had a 1 year warranty on anything that he tested in his report. Does that mean the company will cover all damages or buy the property from you, no; but it is a small disincentive to be a rubber stamp.
And really, the most expensive repairs are on things you can't see. Wiring. Plumbing. Damage to the structure of the house.
Even then I wouldn't be surprised if they tried to weasel out of for example your roof started leaking 2 months after you move in. The inspector will point out that there was no water damage when he inspected it so the issue developed after the inspection.
Also friends, family, coworkers, local real estate agents you trust, etc.
Going into the entire home buying process without preparing adequately is the buyer's first mistake.
Even when they aren't, finding a reputable one who will actually do a good job is extremely difficult. Most are just looking to run through a house as fast as possible and collect their fee. There isn't some directory of problems they've missed. As far as I know there isn't any way to recover costs from them over issues that they missed and caused you to buy a home that you would not have given the problems. This makes them quite prone to being a rubber stamp and yet another fee for home buyers.