Most of us Japanese cannot distinguish ŋ from g
I am afraid this is a hasty generalisation. As a native speaker of Japanese, I can tell [ŋ] from [g] much easier than I tell [l] from [r]. I believe the
distinction between [ŋ] from [g] is rapidly dying out, but it's still premature to say that most people cannot tell them apart.
Looper, [g] and [ŋ] are allophones of /g/. In other words, if you see any of the letters が, ぎ, ぐ, げ, or ご, it can be pronounced either with [ŋ] or [g] at the beginning. Some regional varieties of Japanese, such as Tōhoku dialects, tend to use a lot of [ŋ], while others use [g] more.
The rules for realising /g/ in the Standard Japanese are:
1. It is always [g] at word-initial positions. E.g., がっこう (学校) is [gakkoː]
2. Excepting loan words from European languages and Sino-Japanese compound words, it is realised as [ŋ] in intervocalic positions. E.g., さんがい (三階) is [saŋŋai].
I assume your "が" is the nominative marker. Since it always follows a word, the /g/ is never at the word-initial position. が does not make compounds, either. It is pronounced [ŋa] normatively.