Why does my throat start closing up and tighten at random times during singing. It feels like air trying to squeeze through a small hole when it happens?

It means that you are not breathing properly while singing. While singing breath with your stomach. Vocal exercises will help. Posture is also very important.

Do:

Be relaxed and natural

Keep your movements fluid

Keep your chin level

Keep your knees loose

Keep your head up

Keep your shoulders sloping and relaxed

Keep your toes pointed forward with your weight on heels and soles

Keep the front of your neck loose--don't stretch it

Keep abdominal muscles relaxed

Keep your back muscles relaxed

Don't:

Drop or hunch your shoulders

Move stiffly or jerkily

Tuck your chin in when singing low notes

Stretch your head upwards when singing high notes

Strain your abdominal muscles