Don't we lose image details in doing so?
We do lose information by resizing the images. You have to consider an important fact about scaling down images. If you want to find an exact position of an object in the input image, you usually need the real size image, with real aspect ratio. In papers like YOLO which attempts to find exact positions, it uses a down-scale value which is near $4$ I guess. This is acceptable because it will introduce at least $\pm2$ pixel errors. In classification tasks, you can usually see images with $224\times224$ dimensions. It is an acceptable dimension that keeps the main structure of the images. You should be aware that it does not keep the aspect ratio but it is not a big problem due to resizing images while test time.
What would be the consequences if we scale down the image to some other higher resolution such as 512*512 or 1024*1024 or some other.
Something like the former case is already done in
YOLO and other precise object localisation and annotation tasks. Its consequence is that the number of computation and training parameters increase significantly. Increasing the number of training parameters has a prominent side effect. If you have a large number of parameters, you have to increase the number of training examples, otherwise, the chance of overfitting will be high, albeit you use regularisation techniques.
Can we feed the network without 1:1 square images?
Yes, you can. As I've already mentioned, depending on your task you can preserve the aspect ratio or not. In this case, you have to be aware of your convolutional operations if you set them to VALID operations. Because the dimension with the smaller number of entries will be finished sooner. Consequently, you have to choose the windows size and the type of convolutions wisely.