Yes, 6000x4000 pixels is the original full 3:2 size of the D3300 cameras images. If it was cropped in Lightroom, you can always undo the crop and start over, and crop as you wish for a current print paper size. Having more pixels around the subject should be a big help to match the paper size. Right click on the image in the Lightroom crop tool, and you can set 4:5, which will be right for 16x20 paper. Then adjust the crop borders to be where you want them.
Dpi... So many ifs and buts. Cropping should not change the dpi value number stored in the JPG image file (which is a meaningless arbitrary number from the camera, which has no clue what size you might wish to print it).
But if a raw image, or if using the Lightroom raw editor on JPG files, dpi will be whatever number you told the Lightroom settings to store at any output.
You said 240 dpi, and I think that is Adobes original default, until we specify something else.
Click that link below the image in the raw editor, where it says sRGB.... ppi, and you can change the dpi number to any other number. This number has NO EFFECT on your image pixels, but that number will be stored in the output JPG file
But what matters is the image size (dimensioned in pixels) and the paper size (dimensioned in inches). Then pixels / inches = pixels per inch, which I call dpi.
So yes, because cropping reduces the pixel dimensions, so it does reduce dpi. Not speaking of the stored dpi number, but instead defined this way:
Image size is 6000x4000. If you tell Lightroom to set dpi to 300 dpi, then it could print 6000 px / 300 dpi = 20 inches size (in that dimension, if it were printed at 300 dpi).
But if you crop it, or resample it, either way to be say 3000x2000 pixels, and set the same 300 dpi, then it will print 3000 px / 300 dpi = 10 inches (if printed at 300 dpi).
Which you can ignore, and tell the photo finisher company you want an 8x12 print. Then they will print it 3000 px / 12 inches = 250 dpi.
If you gave them the original 6000x4000 px image and asked for 8x12, this computes 6000/12 inches = 500 dpi, but they won't do that, 500 dpi being pointless, they will first resample it smaller to an appropriate size. Their equipment probably does 250 dpi.
I don't mean to double talk to you, but this is how it is. Dpi is how the pixels are to be distributed over the paper size. The dpi number stored in the JPG file will not be used by the photo finisher company... if you ask for 8x10, then 8x10 is what you get (your dpi number is necessarily ignored).
Your goal is to be sure you give them enough pixels to allow that to be good quality.