"få" means that someone gives it to you
"hämta" means that you pick it up yourself
So the difference between, 'Jag ska få det från honom och Jag ska hämta det från honom' would be like I will be receiving the said thing from him and I will pick that up from him.
So, a more accurate translation would be "I can go get the mustard" or "I can fetch the mustard." Is this correct?
English has "get" to mean both "receive" and "obtain." Most other languages dont have a word that means both.
Could hämta mean somehting like download? (I know that not in this sentence but I've seen it...)
Sure, download can be "hämta" or ladda ner". The "Downloads" folder on a Mac is called "Hämtade filer" by the way.
Is it still the case? To my ears, it sounds like the link posted by @HelenCarlsson now - but my ears are foreign. 2016-06-11.
Yes, the new TTS is actually slightly off as well. It should be pronounced "sènapen" (with a grave accent).
You can read about the pitch accent here https://en.wikipedia.org/wiki/Pitch_accent#Norwegian_and_Swedish
or watch this video