Dodging a heat seeker as Mothra is really easy, as you can fly around the screen and he isn't particularly tall. However, Godzilla is rather big and can't fly, but there is a way to avoid them.
When one appears clear the path directly in front of you with attacks, then back up as far as you can and start jumping, which causes the heat seeker to travel high. When it is close quickly drop to the ground, crouch and move forward. By the time the heat seeker drops low enough you should be past it already.
If that looks a bit too risky then you can wait for the seeker to get close, then use your breath attack and move forward. You'll take damage, but since Godzilla doesn't get knocked back while using this attack then you won't get trapped by it.
Enter the password DESTR0Y ALL MONSTERS (with a zero, not the letter "O", in the first word) to face only the bosses/final monsters.
Enter the password M0NSTER 0 (with zeroes, not letter "O").
Y4RKAQMDR1 G6S4XP-Mars (last hex)
4J62STRMH9 CWU58L-Planet X
7JRNOO4NNU 9WLJT3-Saturn (last hex)
X STAR-Start on Planet X
To play this hidden level, go to the Password screen, but leave the password blank and hit End.
To access a sound test, enter the password S0UND (with a zero, not an "O").
Enter the password START T0 END (with a zero, not the letter "O").