Minimal Effect: Armor and Weapons not required
You can make viable builds as a Druid (and as other classes), without requiring the use of manufactured weapons or armor.
AC Options
Barkskin can help give your AC a boost (up to 16, but it is concentration), which isn't terrible. You're not going to be a Tank, but your Tankiness as a Druid is more in the HP soaking with Wild Shapes than in AC. In comparison, studded leather with a max Dex of +5 would still only give you AC 17. IN addition, when you WIld Shape, you will not have the benefit of your armor and have only the AC from the Wild Shape beast. Any spells supplementing that would carry over, so it's actually a plus for a build like this!
You can also have Absorb Elements prepared to help mitigate some damage types.
As you reach higher spell levels, Stoneskin is another option to mitigate damage.
Finally, the Investiture spells from Xanathar's give further help in mitigating damage and increasing offensive capabilities.
There are also Feats that may be worth it (Magic Initiate) that would let you take a 1st level spell (Mage Armor!) for a non-concentration AC of 13+Dex.
Weapon Options
As you stated in your initial question, Primal Savagery is a great cantrip for you. There are other options as well:
Shillelegh is a great early cantrip if you feel that a quarterstaff(doubling as your focus) is reasonable.
There are a lot of other cantrip and spell options for Druid that let you attack without the need for a manufactured weapon.
Common Gear
Getting by without some necessary items (like a pack to carry things, food, water, etc.) may be slightly more difficult.
You can use Goodberry to bypass your food (and maybe water) needs. But standard adventuring gear may be an issue. You'll need a lightsource if you don't have darkvision (although Produce Flame can give you light in a pinch as a cantrip)
Components/Focus
The biggest thing you'll have to figure out is spellcasting focus and/or component pouch as well as how you carry other things (but it sounds like that's not an issue since you don't want things.) If you decide to forgo the focus or the pouch, you may be limiting yourself to verbal/somatic only spells. If so, be careful in choosing your prepared spells for those with only V,S - but work with your DM and think about how/if you'd like to carry a focus or pouch.
Supplementing with Magic Items
There are a LOT of magic items that could help supplement a build like this. Ranging from the Ring/Cloak of Protection to wands/staves for Druids. My list above was really about how to make a build like this without resorting to items, but those are definitely options for you to review in the DMG if your DM is willing to make them available to you.
Going Commando
Going without arms and armor really is not a problem. You may have some limitations you wouldn't, but they wouldn't be immense. Play your flavor, you can do it well and not feel like you are creating too many difficulties that will stop you from enjoying your character.
Best Answer
It depends on how realistic a game you're running.
In real life, when you pour oil on a sword, the oil basically just runs down the sword, and only a very thin coating will remain on the blade. This coating will either evaporate before you can do anything useful with it, or burn for a moment or two before burning away. In the best case, you get a bunch of burning oil on your hands.
Now, you might be running a more cinematic game, where you want people to be able to pour oil onto their swords and have it be an effective tactic. If that's the case, then you would probably use the same rules that a torch does, and have the sword do 1 point of fire damage in addition to its normal damage. How long this buff will last depends entirely on how cinematic the game is, but it probably shouldn't last more than a few rounds in anything but the most ridiculous of game worlds.
Burning weapons in real life are almost exclusively based on projected fire, or blunt impact of a flaming object. It's not really possible to light a sharp blade on fire without either providing a continual source of flame (like a flamethrower), or using a material that is inherently flammable, which would screw up the weapon's efficacy as a blade.
It's a little bit easier to use a non-bladed weapon as a burning weapon, but it's more likely to damage the weapon that way. The same point about the oil running off applies: even materials that would normally absorb the oil, like wood, will generally be made non-porous when crafted into a weapon. A wooden club that gets moldy when you use it in the rain isn't so useful in the long term. However, if you soak a rag in oil, tie it around the business end of a blunt weapon, and light it on fire, then that weapon can almost certainly be used to cause fire damage in the same way a torch can.
If you soak someone in oil through a thrown flask and then light them on fire, then they will probably take some damage from that fire until they can put it out. If they're wearing flammable clothing, then the fire will likely last until it is actively put out. If they're not, then it will only last a few rounds. The DMG doesn't give a specific number for how much damage fire does, but your instinct of 5 damage per round sounds reasonable.