Is it a good idea to determine coordinates of object on images with regression algorithms? I am doing doing one of getting started competitions on kaggle, the one which requires you to find coordinates of different objects on images.

I know that the usual method for this is to use sliding window and predict whether some piece of image is that object. But I wonder, what if I just use values of all pixels as features and try to teach a regression algorithm how to predict the coordinates of the object on that image? What kind of regression algorithm would be best?
It is possible to train a neural network to solve a regression problem.
See facial keypoints challenge
You should note that training a network for regression is harder then training towards a classification problem.
The network is more prone to unnormalised, different deviation values per feature, weight initialisation etc.
Try following the tutorial and then you'll have a better grasp on using ConvNets for regression.

Good luck!