I'm starting backwards on this answer. From the final question.
Exporting to pdf
When you export a pdf format for print, you should use a specific subformat for example the X family X-1a.
This format excude unnesesary features and process the file so it only have the data you need for print. This are old formats but work well. This dosen't alow transparencies and convert all color mode to cmyk by default.
Grayscale images will be converted using the K channel and spot colors, duotone and rgb colors will be converted to CMYK. I don't rely 100% on this automatic conversion becouse I want to control the flatening of transparency and overprint myself, but they do their job. (See aditional foot note on the bottom)
This process also any rgb external an linked file when you compile a pdf. So if you had an external rgb file will be inside the pdf in CMYK aswell.
If you need spot colors however you need to be careful about RGB files, becouse as you don't want to convert the spot colors to CMYK, the RGB files won't be processed either.
When you import a bitmap file (import, not link it) The image is now part of the document. The original file format is "lost". It is not important anymore. So it dosen't matter if you had a bmp, jpg, png, flat psd or whatever. Just be careful about the resolution.
On old versions of ilustrator any RGB file was converted when imported into the main color mode configured when creating the Ilustrator file. CMYK in this case.
I am not sure about the new versions. But It is not a problem to have an RGB file inside if your output is a controled pdf file. Sometimes is better to have an original RGB file.
One final note. You can always open a pdf again in ilustrator and see what is inside, how the raster images were processed etc. If I have any doubt I just open my exported file and poke inside.
There are some aspects that need to be considered when doing this RGB->CMYK conversions, and you need to be sure your export handle this the way you need it to be. Is the conversion using the right color profile? The black produced is the one you need? Do the gradients look right, and have a smooth color transition? Do you need "pure" colors? Do you need another kind of color conversion, like perceptual or colorimetric? etc.
There are a lot of things to consider when making a color mode conversion.