What's wrong with my gear shifter?
This appears to be an EZ-Fire type shifter.  Shimano or copy-cat. One model number is ST-EF29.  That spring is misaligned and you'll need to get that seated properly. Then I suggest a good dousing with WD-40 or equivalent and exercising the shifter repeatedly. You may need to use a toothpick or the like to scrape out old grease and dirt.  After cleaning, follow up with some sort of wet lube and work the shifters again. I've linked to a video showing some work being done on a very similar shifter to yours. 
Typically I don't find these types of shifters "broken" but rather gummed up or pieces disconnected like your spring. It's common for the junction to the gear indicator to be disconnected and/ or the little plastic connection broken. This doesn't affect the actual shifting just the gear indicator.
You have a good chance of getting this working sans the "dodginess."