Now based on the 37.5 cm of fish I can fit on my tank (I think I can have that much) I have 13.5 cm of fish left I can fit in. Is this true?
I'm assuming you're using a stocking calculator to come up with those figures, making that plan highly inaccurate. There's no rule of thumb when it comes to stocking fish in an aquarium.
If so can someone please recommend some fish to fill that space?
I believe that your aquarium would be well suited to 15-20 small fish (tetras and rasboras) and a single gourami. A single bristlenose catfish (not a pleco) would be fine in your aquarium but half a dozen otto catfish or small corydoras would be much better suited.
Also I think that the Gourami and Tetras prefer a well planted tank, do these have to be live plants? Because its my first tank I would prefer to go with plastic plants, but I am willing to give this a shot if I have too.
Live plants aren't as hard to grow as they seem, you just need to ensure you have the correct plants for the job. Often plants sold in pet stores aren't true aquatics and will die within a week or two. Look on here or TradeMe for plants such as Java fern, Java moss, Xmas moss, hygrophila polysperma, crypts and indian fern. These plants are all really easy to care for.
I think I need to keep the tank at 25 degrees celsius? Is that correct?
The fish you've mentioned have a temperature range of 23-27 degrees.
Also is the hardness / softness of water in Nelson suitable?
What is the hardness of Nelson water?
And finally are those fish compatible?
The fish mentioned are compatible, unfortunately we do not have cherry shrimp available in NZ and our native equivalent is not suited to tropical aquariums.