What do I need in order to complete hard difficulty dungeons?

I was able to complete hard mode on the dungeons, dailies, and events once I got a team with average level in the low 30's.

Composition wise, I try to use the following party makeup as much as possible:
Black Magic/Spellblade
White Magic
ATK
ATK
ATK

Spellblades are nice because they usually have good ATK, but they can also use the strike abilities to exploit elemental weaknesses on enemies.

It is always nice to have as many abilities as possible, but here are some you will definitely want to create and hone:
Double Cut (probably 2 or 3)
Boost
Cura
Curaga
Fira/Firaga
Blizzara/Blizzaga
Thundara/Thundaga
Armor/Power/Magic/Mental Breaks
Bio/Biora
Thunder/Blizzard/Fire/Drain Strike

You mentioned that you already knew about record synergy on gear, but it is so powerful, it deserves another mention. Farming up 1* and 2* gear to combine/upgrade to ++ gear can greatly improve a team. My lvl 12 Wakka has been holding his own in the Celes event hard mode due to record synergy gear.

For the daily dungeons it was around level 30 before I could clear Hard mode. I still lose sometimes.

What I have found most useful so far is:

• Even one piece of 5* gear makes a huge difference. Save all your mythril for the rare drop events, and wait until you get to 50 mythril so you can get 11 pulls for the price of 10. I got the Ice Whip for Rydia this way and her damage output went crazy, to the point where she could autoattack for more than front-line fighters
• Have a secondary healer, even if all you can give that healer is Cure. If you only have one healer and he or she dies, you will probably lose. Additionally, you will frequently need 2 healers to keep up with the enemies' damage. One heal per round is frequently not enough. I typically use Celes or Terra as the secondary healer since they can wear good armor (improving survivability) and can also do good damage from the front row when you don't need the extra healing.
• Use non-targeted heals most of the time. It is a little faster since it's one click instead of two, but the real benefit is that the target won't be chosen until the spell is actually cast. This is very helpful. For example, if one of your party members gets severely hurt while your heal is in the queue, the heal will go to the most injured person (and you probably wouldn't have chosen that one as the target since he or she may have been at or near 100% when you chose to heal)
• Buffing your party gives more value than debuffing the enemy because the buff will frequently last for 2 or 3 enemy groups. For example, I like to use quickness on Cloud to get a few extra actions over the course of the battle.
• Use your strong abilities early and often. Since daily dungeons have only 9 enemy groups and no boss, it's not as important to save your abilities for the end. You will probably get your limit breaks by the end anyway.

As for Expert mode dungeons, the "difficulty" number serves as an estimate for where you should be. The difficulty levels seems to be accurate relative to each other, but not so much in relation to your party's level (my current wall is around difficult 50 with a mid- to high- 30s party).
A good rule of thumb I use for those is:

• If I am able to clear the trash fights with autoattack only and stay near full health due to potions, I should be able to handle the boss easily and get a Master rating.
• If I have to turn autoattack off to heal once or twice, I will probably be able to beat the boss but a Master rating is unlikely.
• If I have to turn autoattack off to choose more powerful attacks, I might be able to beat the boss but will probably lose.

In event dungeons, notice whether or not the dungeon has a boss. If it does not you can be more aggressive with your abilities (as if you were in a daily dungeon)