grayscale

It doesn't really matter because the PDF will be RIP'd at the print shop.  In the RIP they choose Grayscale.  However, if you use only Black elements in your original file, eventhough it is a CMYK file, it will be considered Grayscale. Use "Leave Color Unchanged" in your PDF settings.  Always request a printed contract proof before approving the entire print run.

No, not true gray as a colour mode, only RGB or CMYK. However as Tromboniator stated converting the colour to Grayscale will ensure that the output value is 0c0m0y?k in the PDF. Just as InDesign does not have a true gray mode, the K channel of CMYK serves that purpose.

Depending on the downstream workflow this may or may not be acceptable or you may need to inform them to only use the K channel and not to worry about the CMY triggering preflight warnings etc.

If you truly need the objects specified as gray then you will then need to use Acrobat Pro (Preflight profiles or Convert Colors command) and or another PDF tool such as the PitStop Pro plug-in to remap or convert to create a true, single channel gray element. You will need to consider your colour management settings to convert from CMYK to Gray without changing input/output % values (you may or may not care about slight shifts of say 4% or so).

Select All, go to Edit > Edit Colors > Convert to Grayscale, Save As > Adobe PDF.

Peter