What determines whether a character can use a weapon

Actually, Critical Hits are subject to the RNG, just like To-Hit and Dodge.

Your base chance to land a critical hit is [(Skill / 2) + Class Critical Bonus]. So every 2 points of Skill is worth an extra 1% chance to crit.

This is further modified by:

• Your weapon's critical rate (Weapons with the Killer prefix are especially good for this)
• Your skill in the chosen weapon (0% at anything less than S Rank, 5% for S rank)
• Supports/proximity/Pair Up combinations that directly increase critical chance.
• Skills that increase your critical chance.
• The enemies Luck stat will directly reduce this percentage. (ie. 10 Luck will drop chance to crit by 10%).

The crit rate displayed on the character's info page takes all these bonuses into account, except the opponent's Luck stat. So the simple way for you to figure out your chance to crit is to take that stat, and subtract the target's Luck. You then have a percentage chance to crit.

Most of this info is pulled from http://fireemblem.wikia.com/wiki/Critical_hit#Calculations. It's not specific to Awakening, but from the data I've observed playing, it's spot on.

In order for you to actually GET a critical hit, though, you need the RNG to come through for you twice:

1. Once for the actual To-Hit chance (can't get a critical hit if you miss).
2. And once for the critical hit itself.

And on the (at the beginning) really low chance you actually score a critical hit, congratulations! Critical Hits do triple damage, which tends to be a one-hit kill.

You are right that DDG is short for Dodge. It's similar but different then Avoid.

Avoid is used to avoid/dodge Normal hits, like you stated correctly. With higher AVO you have a lower chance to get hit at all.

Dodge is used to avoid/dodge Critical hits. So with an increased Dodge, you have less chance to get critical hitted, while NOT reducing the chance to get hit at all.