Can printers print from InDesign packages directly or only from PDF?

"I am concerned about the image quality."

You CAN create a PDF with no downsampling and no compression, but you will be sending the images at full value into the PDF - the drawback being you will have massive PDFs ...for no real reason.

The standard print "standard" is to make sure you have a minimum 300 ppi for a roughly 2-to-1 ratio to the printed line screen (e.g. 150-175lpi halftone screen). Any high-end PDF preset use these settings as "standard"). Any more than that is overkill*

*There are exceptions to that, of course: If you use text in your files (e.g. when someone creates an ad in PS and sends the printer a 300ppi file, that is NOT enough for crisp rendering of small text at print stage. There I would recommend a minimum of 600 ppi export with NO antialiasing.

"Can I ask my printer to print from the InDesign package directly"

No. That is not a workflow in the industry anymore. Even if you found one that accepts native files (I can assure you we don't and actually refuse such jobs), any commercial printer will create PDFs anyway (or another export format appropriate for their RIP) which are then submitted to their RIPs/imposition workstations. Even then, you are at the mercy of THEIR settings.

Keep your PSD files in RGB color mode when you place them in InDesign (the text and colour panels can be in CMYK). Ask the printer for a PDF spec, if you can't get one select PDF/X-4 from the InDesign Adobe PDF presets. In the subsequent Export Adobe PDF dialogue box, select Export as Pages (not Spreads), and under the Marks and Bleeds tab select Crop Marks and tick Use Document Bleed settings.

Another thing: Don't print a PDF, export as Print PDF.

Send never open INDD to the printer as following problems can occur:

1. The printer must have the same version to avoid recomposing. Best on the same OS.
2. The printer must be able to handle InDesign properly as well all the others applications he gets files from.
3. Does the printer use the same dictionary as you use? Are you sure that he is not using a local user dictionary with different entries which might recompositions and hyphenations.

There are some other reasons. Often prnter open linked files, like ai. If they do so they destroy the functionality if linked assets or fonts are not avalable.

If you use copyright fonts you are not allowed to pass them on to the printer. The printer has to buy a license and have to use the very same version as the InDesign user used. Otherwise a recomposition could happen.

Any time one prints from InDesign, one takes the chance of change in the file. A PDF is better if built properly. Depending on your InDesign print settings, InDesign may be using Optimized Subsampling. 

If you create a PDF/X-4, you can disable downsampling and compression when creating your PDF. However, depending on the printing method, having too much resolution is not always better and can "soften" the image.