You can really assign a number to transparency. there may be different glasses with different degrees of transparency which can be measured scientifically. But when you just say "high" and don't mention any numbers then it could be 70 out of 100 or 80 ou of 100 or 90 out of 100. it is not precise and for this reason we say "a high degree". We say "a" because we only know that it is high and we don't know the exact number.
Here you should read "high" and "degree" together , you should read "high degree" as a phrase. We are not talking about numbers anymore. "high degree" here just means "importance" , "greatness", ... , so you don't need to say "a" or "the" before it.
About your second question. Tell us why you want or suggest to change it? Do you think the translation is wrong? Why?
About your third question. It's written in my dictionary that you can say it with "a" or without "a".
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