The questions you are asking tell me that you have not read a "reloading how to book" from cover to cover ... ABC's or Reloading would be a gret place to start.
With that out of the way ...
Yes, the new bullets will "fall" into a fired shell. The neck of the shell expands when it is fired.
When you were told you didn't need to resize the cases, I believe you mis-understood the person telling you this. You don't need to Full Length Resize brass that is firedin your rifle. But ... you will ALWAYS need to neck size your brass. Read your reloading manual to understand the difference between them, and the reasons why a person would choose one over the other ... an entire thread could be devoted to just that.
Yes, .308 diameter bullets are correct for your 30-06. Look at the page in your reloading book that you are getting your loads from ... it will tell you the bullet diamter, along with the weightof each bullet and the powder options available.
You do not "flare" the mouth of a rifle case.
Not all rifle loads require crimping. Again, a reloading "how to" book will explain when and why you need them, and even when/why you can use them even if you don't need them.
Lube: you lube the outside of the case before resizing it.
Cleaning ... this is pretty much an :up to you" thing ... some guys prefer to clean them before resizing and depriming, to eliminate the need to manually remove teh cleaning medium form teh flash hole ... other guys like the fact that the media can clean the primer pocket if the primer has been removed during the resizing process. Myself, I clean first, then resize and deprime, then use a pocket reamer to clean the worst fo the crud from the primer pocket.
You didn't mention case length ... Are you mearuing it before starting to reload? are you deburring the case mouth after trimming it to the proper length?
Again, I cant' say this enough .... based on the questions you ahve asked, you really need to get a quality reloading book and read it thoroughly. ALL of your questions would have been answered and prevented had you done that before you started. When you know what you are doing (well informed), reloading is a fairly simple and safe process - when you don't it's still simple, but can lead to catastrophic results.
edit: Where are you located? I'm just north of Toronto, and for the price of a Timmies, I'd be happy to spend an hour or so with you and give you a hands on "intro".