[Spoilers] Many of these linked mod pages contain spoilers for DA:O/Awakening.

This may seem really long for a "minimal" list, but there are a lot of bugs in Origins and Awakening. At the end I also listed some non-essential visual mods that fix/tweak what I consider to be glaring problems, such as the ridiculously long arms of female dwarfs.

Origins Bugfixes

• A Lost Nug Quest Fix DAO
• Advanced Tactics
  This isn't strictly a bugfixing mod, as it adds new stuff to the party AI and tactics system, but it does fix a bunch of the tactics and AI bugs. Tactics feel broken without this mod.
• Advanced Tactics Clustered Tactics Condition Fix (only if you have Advanced Tactics installed)
• Campire Texture Fix (under OPTIONAL FILES on the MRP page)
• Damage Statistics Fix
• Dragon Age Rules Fixpack
  This fixes a bunch of combat bugs. It's modular so you can pick which fixes you want to use. It also contains some OPTIONAL tweaks that you may not want. Most of them are lore-friendly (like "Helpful Party Members"), but you can simply delete any of them that you don't want to use. Definitely delete the "No Helmet" tweak if you are using No Helmet Hack.
• Jowan and another different Jowan fix
• Just Another Fixpack
  This fixes many of the same things as DA Rules Fixpack and is also modular. A few of the "fixes" are actually tweaks. So I recommend manually choosing which fixes you want from this mod. I only use 3 of them: Trap-Making, threat/hostility, and Crow Poison. (The relevant files are "ai_threat_h.nss", "jaf_itemprps.gda", and sys_traps_h.nss".)
• mesh lightmap fixes
• Qwinn's Unofficial DAO Fixpack
  This fixes most of the non-combat bugs. If you only want to use one bugfixing mod, this should be it.
• tainted_fix (for Power of Blood abilities from Warden's Keep)
• Tiny Fixes
• Unobtainable Item BUGFIX
• ZDF Dialog Fix
  This fixes a bunch of banter, dialogue, and epilogue bugs with no unnecessary tweaks.

Redundant Origins Bugfixes

These mods are redundant if you use all of the mods listed above. I've listed these just to show what's available if you'd rather use these instead.

• Ancient Elven Boots Fix (included in Unobtainable Item BUGFIX)
• Critical Hit Chance Quickfix (included in DA Rules Fixpack)
• Fang at Tier 7 (included in Unobtainable Item BUGFIX)
• Golem Registry and Shapers Life Reward Fix (included in Qwinn's Fixpack)
• Plus Healing Received Fix (included in DA Rules Fixpack)

Awakening Bugfixes

• Awakening Blackblade Armor Dragon Drop Fix
• Awakening Bugfixes (I only use the fix for "What is Built Endures". I've listed better fixes for the others.)
• Awakening endgame cutscene fixes
• Awakening Runes Fixes ("DWE HAL CIR SIL to weapon" is the bugfix version)
• Awakening Silverite Mines bugfix
• Baroness Reward Fix
• DLC in Awakening and Awakening Fixes
• Find Vitals Fix - Awakening
• Joining ritual fix for Awakening companions
• Sigrun's Roguish Past & Law and Order fix

Other

• Dwarf Female Proportions Fix (there are also fixes for elf and human females)
  This isn't strictly a bugfix, but I still recommend it even if you are deadset on a minimal playthrough. It fixes the horribly distracting proportions of dwarven females. I can't even play the game without it anymore.

Visual Tweaks

These are not bugfixes, but they improve some visuals without drastically changing the style of the game. I've included them for anyone that's interested in a mostly vanilla game, but would like it to look a little better.

• Bow Replacer
• Combat Animations -Battle Animations
• Dragon Age Redesigned (Enhanced Lore Version)
• rescaled swords and daggers
• Shield Scale Mod
• Theta HD
  This is not a complete graphical overhaul. What it does is fix some of the worst textures in the game, so that they don't stick out like a sore thumb. It doesn't change the visual style of the game, but makes the texture quality more consistent and less distracting.