It depends how many fish were in the tank when the previous owner had it. Fish make ammonia as their version of urine. Bacteria grow in the filter, and on other surfaces, which use this ammonia as food. But there will only ever be the amount of bacteria that there is enough food for. So if the previous owner had 20 harlequin sized fish, there will be the right amount of bacteria to eat all the ammonia made by 20 harlequins, in which case you could get more fish straight away. If he had 40 harlequin sized fish, you could get even more fish. But if he had just 3 harlequin sized fish, you have too many fish at the moment for the amount of bacteria. But this does not mean you can never have more fish. These bacteria will multiply if there is more ammonia than the current number can eat, you just have to take it slowly and not get too many fish at one go. This is one reason for getting a test kit so you can monitor the levels of ammonia and nitrite whenever you get more fish to make sure the bacteria are coping.
The ammonia eating bacteria turn the ammonia into nitrite, and there is another type of bacteria which eat that and turn it into nitrate. Both ammonia and nitrite should never go above zero, but they will if you get a lot more fish than the bacteria can cope with.
The difference between hard and soft water is the amount of minerals dissolved in the water. Fish from soft water rivers and lakes have evolved to cope with few minerals; they have problems if they are put into water with a lot of minerals. And fish from hard water areas have evolved to cope with lots of minerals and they suffer in water that doesn't have enough minerals. Fish cope better and live longer, healthier lives if the tank water is similar to that in the river or lake they originated in.
So with a GH of 6 (5.9 is near enough 6) virtually all of the fish from South America would be fine, as would most fish from Asia. The fish that would have problems are those from Central America and the Rift Lakes in Africa (Lake Malawi and Tanganika mainly)
The fish on your first list - blue rams and emperor tetras are South American; harlequins and gouramis are Asian. But platies and mollies are Central American.
Look at the fish profiles in the menu at the top of the page. Virtually all the fish in the Tetra section with the word tetra in the name would be suitable from a water point of view, though some of them would need bigger tanks than yours.
Most of the fish in the Barb section would suit your water as would those in the Catfish and Loach sections.
The section to avoid is the livebearer section and some of the fish in the cichlid section (Central American and Rift Lake cichlids). But there are plenty of South American cichlids to chose from.
With softish water, you have a much bigger choice than those with hard water.
With so many fish to choose from, it is hard to draw up a list for you. You may not like the fish I like. The best way to go about it is to visit some local shops and see what they have that catches your eye BUT DON'T BUY ANYTHING. The first rule of fishkeeping is don't trust the shop. Too many of them will sell you the most unsuitable fish just to make a sale. Make notes, then research. Come here and tell us what you like the looks of and between us we can help you make a good selection. Between all the members on here we have experience of a lot of fish