Question: I have a transformer that outputs 14.3V (AC of course) and the current in short-circuit is about 13A (made a beautiful spark when I measured it :) ). Can I guesstimate from here what is the 'normal' current that I can get from this transformer without overheating it? I need 1.3A.

Other data: 

Input voltage: 230V  

Output 1: 14.3V  

Output 2: 18V  

Transformer's iron core: 4.5 x 5.5 x 3 cm.  

It is written on the transformer that it has an internal thermal fuse.  

Transformer name: transformer SF-57E/11B/18A  

From an music player (tape, cd, pickup, radio) with speakers of about (2x) 4-8W.

Answer: 
You need 14.5 V X 1.3 A = 18.9 VA. Look for 20 VA transformers on seller sites. Here is one that I found:

[![enter image description here](https://i.stack.imgur.com/0hIJP.png)](https://i.stack.imgur.com/0hIJP.png)

The one that you have is about that size, so there is a good chance that is will be ok, but you don't have much safety factor in the estimate.
Short-circuit current is not a useful measure.

Instead, pick a tolerance — say, 10% — and gradually increase the load current until the output voltage drops by that amount (e.g., from 14.3 V to 12.9 V). This gives you one estimate of the transformer's capacity.

Pay attention to how the voltage drops as you increase the current. If there's an abrupt "knee", then you want to stop there and back off a bit. This indicates that the transformer core has started to saturate, and you don't want to operate in that region because losses go up dramatically.

Let the transformer run at that load for a while and see how warm it gets — and whether the thermal fuse trips (hopefully, it's self-resetting). If it gets too hot to touch, you need to back off on the load.