Why was a FBI Academy recruit sent to question Hannibal Lecter?

Why? Because Crawford was impressed with her and was giving her a chance to show that she had what it would take to work under him at the BSU

Crawford was giving her a chance to show her value to his team in the Behavioral Science Unit because she had expressed a desire to work in that unit and she had left an impression on Crawford before.

Crawford: A job's come up and I thought about you. Not a job, really. More of an interesting errand. Sit down.

Starling: Yes, sir.

Crawford: I remember you from my seminar at UVA. You grilled me pretty hard, as I recall, on the Bureau's civil rights record in the Hoover years. I gave you an A.

Starling: A-minus, sir.

Crawford: (Reading Starlings file) Double major: psych and criminology. Graduated magna. Summer internships at the Reitzinger Clinic. It says, when you graduate, you wanna work for me in Behavioural Science.

Starling: Yes, very much, sir. Very much.

The book adds some additional details/reasons...

"Yes, well--- Okay, right. I'm glad of the chance, but you have to know I'm wondering--- why me?"

"Mainly because you're available," Crawford said. "I don't expect him to cooperate. He's already refused, but it was through an intermediary--- the director of the hospital. I have to be able to say our qualified examiner went to him and asked him personally. There are reasons that don't concern you. I don't have anybody left in this section to do it."

' "You're jammed--- Buffalo Bill--- and the things in Nevada," Starling said.

"You got it. It's the old story--- not enough warm bodies."

Is this standard practice at the FBI?

I can't speak for the FBI, but yes its common for people to start working for different companies in the form of internships or in an evaluation period before they graduate from school, college, wherever. I would imagine this kind of thing happens at the FBI as well.

In addition to the her qualifications, there are, possibly, other reasons. Remember, Lecter is a psychopath and a genius, and is an expert in psychology. Jack Crawford has done battle with him, and knows that sending his usual experts will be fruitless - Lecter will toy with them, see through their attempts to manipulate information from him, and leave them with nothing but frustration. Starling presents something quite different. She comes in as someone who is inexperienced and, for lack of a better term, innocent about dealing with serial killers. She is more open to exchanging information about herself, even though she is specifically warned not to, which intrigues Lecter.

Jack Crawford: Be very careful with Hannibal Lecter. Dr. Chilton at the asylum will go over all the physical procedures used with him. Do not deviate from them for any reason whatsoever. And you're to tell him nothing personal, Starling. Believe me, you don't want Hannibal Lecter inside your head. Just do your job, but never forget what he is.

Starling is also a young attractive female. If you notice the movie, itself, every scene where they just show her going about her business - walking down the street, jogging on campus, showing up at the funeral, etc. they make a huge point to have many, many males blatantly turn and gawk at her. I think it's pretty clear we're supposed to understand that Starling is supposed to be remarkably attractive.

Dr. Chilton: Crawford is very clever, isn't he, using you?

Clarice Starling: What do you mean, sir?

Dr. Chilton: A pretty young woman to turn him on. I don't believe Lecter's even seen a woman in eight years. And, oh, are you ever his taste. So to speak.

Crawford knows he's not going to be able to best Lecter, intellectually, so he tries to get Lecter to lower those defenses by appealing to whatever semblance of emotional side he might have.

In spite of himself, Lecter is intrigued

Hannibal Lecter: You're very frank, Clarice. I think it would be quite something to know you in private life.

And he enjoys or is entertained by the inter-play between the two -

Starling: I thought you might like your drawings back, Doctor. Just until you get your view.

Lecter: How very thoughtful. Or did Jack Crawford send you for one last wheedle before you're both booted off the case?

Starling: No, I came because I wanted to.

Lecter: People will say we're in love. Anthrax Island. That was an especially nice touch, Clarice. Yours?

Starling: Yes.

Lecter: Yeah.

I don't know if it's standard practice to use an intern/student in this manner, with this level of responsibility, but Crawford felt like he had nothing to lose, really. It's common for the police and FBI to engage in psychological manipulation, for sure, though.