I am confused by when to disenchant something. I have disenchanted a sword that did 5 points of fire damage. Will I be able to eventually increase the amount of fire damage higher than 5 points as I get better at enchanting + higher level destruction + high level soul gem? Or should I have waited until I found a weapon that did 20 points of fire damage to disenchant.

I am worried that since you can only disenchant an enchantment just once I have screwed myself.

You learned the "fire damage" - not the "5 points". You won't see any magnitude in the list of enchantments, only in the dialog for the current attempt, which will be based on your skill (fortified or otherwise).

I tend to look at the value of the item to be disenchanted rather than what its enchantment is. For example, I'd rather learn the Fire Damage enchantment by disenchanting an Iron Dagger of Embers rather than a Daedric Sword of the Inferno. And quite frankly, I really don't need to know those enchantments until I plan to do a LOT of enchanting. So I have a chest in one of my abodes where I dump everything with an enchantment on it that I don't plan to be using. Along about Level 30 or so, I'll drag all that stuff out (and _immediately_ get MASSIVELY overburdened), stagger over to the Enchanting Table, and start disenchanting the cheapest stuff in the pile.. More expensive stuff that has duplicate enchantment types move over to my Smithing "To Be Improved & Sold" stash.

Just as an observation, if you use enchanted weapons, just simply recharging them with soul gems will significantly boost your Enchantment score. So I don't feel any particular rush to start enchanting stuff, just to get my Enchanting Skill improved.