No. Sanskrit is not gender neutral. Every noun or pronoun in Sanskrit has a gender.
There are three genders in Sanskrit.
पुल्लिङ्गम् pulliṅgam Masculine
स्त्रीलिङ्गम् strīliṅgam Feminine
नपुंसकलिङ्गम् napuṃsakaliṅgam Neuter
How to find the gender of a word in Sanskrit?
The best way is to use the dictionary to find the gender of any word.
Sometimes gender of Sanskrit words is the same as one found in any of the regional languages of India but there's no definitive rule and there could be lots of exceptions.
No. Sanskrit is not gender neutral. Every noun or pronoun in Sanskrit has a gender.