Are there journals that may consider articles of the type "new proof of old results" but the paper is exceedingly short(1-3 pages)?
The answer is certainly "yes", but to identify the best fit for your paper you'll have to decide how different your proof is from previous proofs. It often happens that two proofs look completely different when viewed line-by-line, but still the main intermediate steps in the proofs are essentially the same -- in other words, proofs can use the same strategy but different tactics. If your main contribution is in finding an efficient or elegant way to present some of the same ideas that were used in previous proofs, then you might consider Expositiones Mathematicae or the American Mathematical Monthly. On the other hand, if your proofs use entirely new ideas, then you have done original research and you should submit your paper to a standard journal. In between these two extremes is L'Enseignement Mathématique, which often publishes papers which combine new research with improved exposition of known results.
Most of the time, I would go to any journal that fits the subject (i.e., with an editor that should be interested, and know who to sent the paper to). For example I have had recent success with a 2 to 4 pages (depending on formatting) paper in Israel Journal of Math, which is not especially inclined toward short papers.
Depends on the field and the nature of your derivation. If it gives new insights and uses completely new techniques that might in themselves be of interest, then you might pursue a regular journal. If the derivation is a clever application of an established concept, then you might consider something like the American Journal of Physics, or others with a similar, pedagogic, agenda.