The Purple Dragons are the primary agents for law enforcement in Cormyr. By law, they have the right to question or search any one at any time. Players can expect to be searched at a border outpost whenever they enter or leave Cormyr, but once inside the kingdom they shouldn't be harassed much unless there is reason to believe they're involved in something illegal. As long as the players keep their weapons peace-bonded (unless they have an adventuring charter issued by the Royal Court) and aren't always around trouble (I know, that latter is difficult for players!) and they behave with respect towards the Purple Dragons, there shouldn't be any trouble. Oh, and about adventuring chargers... it's illegal for a group to adventure in Cormyr without one. The group has to register with the authorities and purchase a charter listing each member of the party. Groups of armed people wandering around the countryside are considered to be a potential threat to the security of the Kingdom and thus are required to register and be kept track of.
The Purple Dragons' main interest is the security of Cormyr and the protection of the locals. If the players do something that ends up heading off a threat to the kingdom or that saves the lives of locals, the Purple Dragons should be favorably inclined towards them and give them a little leeway afterwards.
The Thunderstone unit in particular has a reputation for bullying, but I wouldn't overplay that too much. They, like other Purple Dragons, are interested in the security of Cormyr and the protection of the locals. They're just under more stress since they also have to contend with the monsters from the Hullack Forest and the Vast Swamp. I would simply portray them as being more heavy-handed in their upholding of the law (i.e. more forceful in the legitimate exercise of their authority and less tolerant of disobedience or backtalk) and more suspicious of foreigners than is usual for Purple Dragons (foreigners can expect to be searched and questioned whenever something happens) which might include your players; I would not portray them as bullying the Thunderstone locals without good reason nor would I portray them as behaving in criminal ways like extorting or shaking-down people. The Thunderstone Purple dragons are not criminals and the avoid -- if only barely -- exceeding their authority. If your players do some good thing like heading off a threat to the kingdom or saving the lives of locals, then I would portray the Thunderstone Purple Dragons as respecting that and forming the foundation for a more positive relationship with them.
Players that want to raise Hades, cause chaos and generally do what they want will find the Purple Dragons to be stiff at best and more likely harsh in their dealings with the players. The Dragons are the law, remember, and the Kingdom is a civilized nation of law. Cormyr is not a good place for people who openly defy the law. That's just the way it is; people who can't deal with that had best leave the kingdom.
As far as the Purple Dragons not being there until after the trouble's over, well, that's the timeless dilemma of police everywhere in every age, real-life or fantasy. It's plainly and simply not possible for the police to be everywhere and protect everyone constantly, and it's unreasonable for either real people to expect that in real life or the characters to expect that in Cormyr. In order to accomplish that, you'd need a number of police that would put even a totalitarian police state to shame and it'd be even less pleasant to live in than that police state. If the mere presence of a law-enforcing body and their potential involvement isn't enough to deter criminals (like the Fire Knives), then all the law-enforcers can hope to do is catch and punish the criminals after the fact. Your players just have to deal with that too.
And as far as law enforcement goes, don't leave out the War Wizards either. There's a war wizard present at every border outpost and a number of them in most cities. They are not as visible as the Purple Dragons, but they more closely approach having police-state powers. They can (and do) mind-scry people they think are suspicious without warning or even informing them. Remember that it's the law in Cormyr that every wizard or sorcerer of 5th-level or higher (i.e. capable of casting 3rd-level spells or higher) has to register with the War Wizards upon entering the country (this is in addition to having to register as part of a chartered adventuring group), and the War Wizards keep a tight watch on any such mages in the country.
In short, if your players just want to adventure to fight monsters and otherwise not cause trouble for the kingdom for it's people, relations with the Purple Dragons should range from 'not too bad' to distinctly friendly. If they do things that bring trouble to the people or pose a potential threat to the security of the kingdom, they can expect a distinctly unfriendly relationship with the authorities.