Aww he's so cute! I'm really glad that you did adopt him! Don't worry we can help ^_^
As far as color, yes being see through is just fine. The way color works is that the top layer is the iridescent layer and then blue, underneath blue is black, red and then yellow. So literally different colors have different amounts of layers! So that's why many yellow/white Betta's are see through because they have the least amount of layers
As to his bloody fins, nothing that the salt and clean water won't help! I suggest using the salt at 1 teaspoon per gallon as you've already started for 7 days. Do a full water change every day with using the salt (make sure to dissolve first!) and if you've got Stress Coat you can use that as well
After the 7 days, if that red still hasn't gone away you can do another 7 days of the salt/water changes but really just what he needs is good, clean water. So up to 14 days with the salt and then no more salt unless you give him 4 days rest and you can start again if you really want to. But if you see the red go away before that then you can stop the salt and just continue doing water changes. At that point you can do them every other day and not every day so it won't stress him too much but still keeps him in clean water
I'm glad you're using a heater still though. I know many people don't realize, even if the house is heated night and day that the temp still drops at night so the temp swing isn't going to be very healthy for him unless it's like literally 2-3 degree's that's fine but more isn't good.
Any other questions feel free to ask them
I'll be here!