Exporting - what settings should I use?

Question

I am wanting to send pictures to a publishing / printing company who will use them for prints which they will sell to their customers.
I use Lightroom 4.1.
I would appreciate some advice about the settings I should use in the Export window.
Question1: Output sharpening
The pictures have gone through the Develop module and the last thing I did to them there was to sharpen them.
Does this mean that I leave the "Sharpen For:" box unticked in the Export window?
Question 2: Image Sizing
I want to maximise the number of pixels in the exported pictures.
Does this mean that I leave the "resize to fit" option unticked in the Export window?
Question 3: Image Sizing
What do I do about about the figure to be entered into the box labelled "Resolution:"?
Do I ignore it?

Answer 1

davidrw1948,
Why don't you give me the specs of your publisher? Then I could tell you what settings to use.
Sharpening: generally you need two different sharpening procedures for your images. The first one is done in the Develop Module and it eliminates (or reduces) the inherent un-sharpness of digital images. Then for your output - be it print or display on screen - you use another sharpening set ( either in the Print Module or in the Export Dialog). The two sharpening procedures are different and usually one alone does not give satisfactory results.
In your case, the difficulty seems to be that you don't know if your publisher applies sharpness and doesn't want you to apply output sharpness - or, if he doesn't sharpen at all and prints the image just the way you send it to him. For instance Blurb says quite clearly that they don't apply any sharpening and that sharpening is our responsibility. What does your publisher say in that respect?
Resolution: Strictly speaking, there is no such thing as resolution of a digital file. There is only pixel dimension. You can speak of resolution only in relation to print size.
For instance: An image comes out of the camera at 2000 x 3000 pixels. If you don't want to resize this image you'd get a resolution of 300 dpi when you make a print of 6.66" x 10"; but when you make a print of 20" x 30" you would get a resolution of 100 dpi. The formula is: Resolution in dpi = pixel dimension / size of print in inches.
But in the latter example of a print of 20" x 30" with a dpi of 100, the image does not have enough pixels for a satisfying print resolution. 100 dpi is not enough for a good print; a good print requires 240 dpi or - depending on the printer used - 300 dpi (= industry standard for magazines and books), or 360 dpi (high end Epson printers).
So if you want to make a 20" x 30" print from an image that has originally only 2000 x 3000 pixels, you would need to resize the image, i.e. enlarge it, i.e. the software (Lr or Photoshop) has to "invent" new pixels. And when you resize, you have to specify the resolution because for a print of a given size more pixels are needed at higher resolution and less pixels are needed for a lower resolution..
You say "as the printers will want to decide on the resolution that best matches their needs as they will be printing at various sizes and also on different media (as well as photographic paper)" . Question: Do you know that - or do you assume that it is so? Don't assume that printers do things - mostly they just send the image file to the printer, and that's it.
But, maybe this publisher / printer is different.
You say that your printer would print in different sizes. In that case I would make my output file in a size equal to the largest print at a resolution of at least 240 dpi.
If your original file is smaller, you have to resize it.