I think some good questions are asked, and I'll speak up on a few things that I feel are appropriate to discuss on a public board... one thing I'd like to point out, however is that training can vary from year-to-year. While I traveled several years ago, it does depend on who is in the trainer position, and what's going on within the Fraternity (if it's a Convention year, new Council members, etc.). So I guess what I'm saying is that this may not apply currently, but it's how it was done back when.

Also, we did not have any LAs when I traveled (or Graduate Counselors/GCs as they were called!), so I can't speak to that program.
What things are done and gone over in the training? What does the training involve?
Training is extensive and primarily focuses on recruitment during the summer. When I traveled we were there for just over two weeks and it was quite intensive. Now I understand that training is longer. Consultants then go back to IHQ in October (following the busy recruitment times) for leadership development (working with officer teams) and finance training. Our training was conducted by the Director of Chapter Services and other staff members at IHQ, as well as various Council members and VST members. We also worked with other consultant staffs to train on meeting with Greek Advisors, etiquette, program presentation, etc.
How are the LCs/LAs paired up with the schools they are given?
As far as I'm aware, this is done internally based on the input of those who interviewed the LC candidates and reviewed their application and background information (Council and IHQ staff).
How much influence should an LA/LC have on a chapter and its recruitment? Should it be "Whatever you guys want to do" attitude or a "Here's what you WILL do" attitude?
The Consultant is there to do her job by ensuring that the Fraternity's Standing Rules, Governing Documents, and Constitution are upheld - as well as NPC guidelines and local, state/province, and federal laws, of course. If a chapter is not at Quota or Total, then the Consultant has probably been advised by a VST member to implement certain changes – she's not just doing it because she feels like it, or because it was done that way somewhere else. She's working to make the chapter successful - and if something isn't working (the chapter isn't making Quota and Total), then the Consultant can offer ideas and input for improvement. It can be a double-edged sword to have a chapter who is not successful in recruitment keeping things "the way they've always been" just because... when a little change might bring about results. It's also important for all of us to be aware that change doesn't happen overnight! Some changes will take a year or two for benefits to be clearly seen. But in the end, the LC has been thoroughly trained in these areas and oftentimes has been directed by a VST member to initiate change during recruitment.
Hope that helps answer some questions!