Learn Swedish: Brandmannen kom springande mot elden. - The firefighter came running towards the fire.

You are right that Swedish does not use this form to say he is running (that is simply han springer). However, Swedish does use the present participle (words ending in -ande or -ende) to describe the manner in which someone moves. Here, springande acts as an adverb telling us how the firefighter arrived.

How do I know whether to use the -ande or -ende ending to make this form?

It depends on the infinitive form of the verb. Since the infinitive springa ends in an -a, you just add -nde to get springande. If the verb ends in a different vowel, like bo (to live), you add -ende to make boende.

Could I say Brandmannen kom och sprang instead of using springande?

Grammatically, kom och sprang works, but it changes the meaning slightly. It sounds like two separate actions in a sequence: first he arrived, and then he ran. Using kom springande emphasizes that the running was the exact method of his arrival.

Both brandmannen and elden relate to fire. What is the difference between en brand and en eld?