Most likely you need to set View > Display Performance to High Quality Display. By default ID uses a low-res preview. The screen preview has no effect, however, on the quality of the output.
You can, indeed, leave the images as RGB and convert to your CMYK profile during export. That will work for RGB and tagged CMYK objects, but native CMYK or untagged CMYK, which will be treated as being in the document CMYK working space, may exceed your ink limit -- there is nothing that prevents you from defining a color that uses too much ink, even in Photoshop.
The ink limit is set by the CMYK profile when there is a color conversion and not the application. InDesign has an Ink Limit preview in the Separation Preview panel, which shows ink limit violations for the current output.
Here's an AdobeRGB image and a US SheetfedCMYK image placed in a document that is assigned the CMYK US Newsprint SNAP profile. When the Separation Preview's Ink Limit is turned on the areas of the CMYK image over the 238% limit I've set show as red. The RGB image has nothing showing over the ink limit because on export it will be converted to the document US Newsprint SNAP profile, which has a 240% ink limit. In order to get the US Sheetfed CMYK image under the US Newsprint's 240% limit, it needs to be converted to US Newsprint either in Photoshop or on export.
If the document is assigned US Sheetfed Coated, the RGB image shows ink limit problems because US Sheetfed allows 350% total ink, which is what I'll get on export.
Ink Limit indicator turned on and set to 238 with US Newsprint as the document CMYK profile
Ink Limit indicator turned on and set to 238 with US Sheetfed Coated as the document CMYK profile