I read that universities invite candidates and ask them to book a flight. If universities don't specify the amount available, can candidates select any airfare? Should the candidates ask or just book any? 100$ or 1000$, would that matter? And do booking the cheapest flight be a positive sign?
When you're invited for an interview, the university should offer some guidance on how to book travel and what they will reimburse. If for some reason they don't, it's reasonable to ask how you should handle it. By default you should book a reasonably priced economy-class ticket (i.e., given several options don't take one that stands out as much more expensive). It's wise to save a few screenshots of price comparisons in case anyone suggests you should have gotten a much better deal. If all the available flights seem remarkably expensive, for example if you are invited to interview with relatively little notice, then you can always check to make sure they are OK with the expense (but they will presumably say yes if the expense could not reasonably be avoided).
The norm for any job interview is to book an economy ticket and the most reasonable rate (NOT what airline gets you best "points"). It is reasonable to prefer a direct flight and one that works with your personal schedule in terms of departure/arrival times, but even here I would probably do the more inconvenient flight if the difference is over $200. (With some judgment...for instance if you have other interviews or other hard schedule demands, just do the flight that works best time-wise.)
Sometimes last minute travel, especially during peak demand periods (and summer is one), can be quite expensive. So if the ticket is above $1000, I would let them know what is coming. They will very likely say just book it before it goes up more, but it is just nice so it is not a surprise. Otherwise just book it.
Also, you don't need to clear and pre-budget all the details of your rental car/cab or the meal in the airport TGIF. You're a grownup now, applying for a grownup job. You don't need to clear this all ahead of time--it will look unsophisticated if you do. They will probably give you a form to submit expenses (after the meeting)--if not, just send a letter to your point of contact or whatever factotum at the department handles that stuff (if you know name), with a bulleted list of travel expenses: *Cab or POV mileage (@IRS rates) to airport, plus any tolls *Airport parking *Rental car or cabs at destination *Hotel *Meals
Save your receipts--many places expect them to accompany expense reimbursement. (For some, just the list is fine.)