For example, for printing a big size image to an A4 page, its better to resize it manually to exact A4 size, or let printer software set it automatically? is there any difference on sharpness, etc?
There are two aspects of an image "fitting the paper". There is size, and there is shape. Images can be scaled to paper size (with the pixels per inch resolution parameter), but they often must be cropped to match paper shape.
Let's assume some easy numbers, that your image size is 6000 x 4000 pixels. And that you want to print on 6x4 inch paper.
If you simply let the printer driver automatically scale the image to "fit the paper", it will print at 6000 pixels / 6 inches = 1000 pixels per inch. Which is unrealistically too high to be believed possible (for color images). It should still print OK, as best as the printer can do whatever it can do, but it's not a reasonable choice, a waste, you would not want to upload such a huge image. The optimal value for printing color images is considered to be 300 dpi (pixels per inch), which what our human eye might see.
You can first resample the 6000 x 4000 image to be 1800 x 1200 pixels, which can then print 6x4 inches at 300 pixels per inch (1800 pixels / 6 inches = 300 pixels per inch spacing), which 300 pixels per inch is considered optimal, all that our eye can see (at normal view distances).
The usual case is much worse than that though. The usual case is that the image shape is not the same as the paper shape. 4x6, 5x7, 8x10 papers are all different shapes, and your image shape might not match any of them. So of course, that can't work right until corrected. If the image and paper shapes were different (for example, a 4:3 image to go onto 3:2 or 4:5 paper), then some part of the image would be cut off past the edge of the paper. This is of course a choice, either to wait for the surprise, or you can chose to first crop it to actually fit the planned paper shape. This is NOT about the printer, it is about the size of the selected print paper. Your doing the cropping yourself lets you choose what area of the image will be lost off of the paper.
This example image shape is 6000:4000, which ratio reduces to 3:2, and it is same as the 6x4 inch paper shape (aspect ratio), and so does not need cropping to fit the paper shape. However compact cameras and phones take images that are 4:3 shape (meaning, they don't already match any common paper shape). DSR camera images are 3:2, but they don't match 5x7 or 8x10 paper.
The typical case is that camera images usually do not fit the paper shape, until they are cropped to match shape. If you send your image to the one hour printing place, they will resample its size first to fit the paper size you specify (typically 250 dpi is common), but the automatic printer does any cropping (when it simply does not match the paper shape).
Printing at home, the printer will use the pixels per inch resolution number (if that number is in the image data). Or it will offer to scale to the paper size. But any automatic cropping is simply done by the paper size, so some of it probably may fall outside the paper size borders.
This is about the least that we must know about printing digital images. More about doing this necessary "resize" at https://www.scantips.com/lights/resize.html
In my experience, you probably will not be able to see the difference between a shot prepared specially for the printer and an image that is scaled down by the printer, but my experience is limited and included some pretty nice quality printers. What I suggest is that you test it yourself. Are you really going to take the word of a few strangers on the internet over whatever you can see with your own eyes? Scale one image to the exact ideal size for the printer. Print the original (larger) version without pre-scaling and compare the results. Which looks better? Can you even tell the difference? Either way, you have a better answer for your situation than we could possibly give you without even knowing the brand or model of printer or other software that might be involved.
There is no way to tell what different software does. So there is no answer to this question.
When printing a large image to a smaller size it could downsample or just use the pixels as they are, but you probably will not notice any difference.
When scaling up a small image you could note a difference, if it made some kind of bicubic resampling or just printing bigger pixels.