Does casting dispel magic on a plain +1 sword make it no longer +1? Also does dispel magic work on a dragon slayer longsword such that it is no longer +1 and no longer does extra damage against dragons?
No, it doesn't.
Dispel magic says that "any spell of 3rd level or lower on the target ends." (PHB p.234, followed by some provisos about higher levels. Emphasis mine.)
Your +1 sword is not a mundane sword currently under the effect of a +1 to weapons spell, it's just a magical item. A spell may have been used in its creation (see DMG pp.128-9), but there's no currently-active spell you can point to whose effect you'd be dispelling.
No, but Antimagic Field does
Magic Items. The properties and powers of magic items are suppressed in the sphere. For example, a + 1 longsword in the sphere functions as a nonmagical longsword.
It's important to note that antimagic field is not permanent, and any magical properties return outside of the field. It doesn't make sense that a relatively low-level spell, such as dispel magic, would be more powerful and permanent than the high-level antimagic field.
Ah touche! Dispel Magic does indeed affect Magic Items, given that you can target objects. Which is what led me to this thread... There is no duration.
My interpretation of RAW is that it could destroy a magic item if the spell effect is less than 3rd level. If the spell effect is of unknown level, I assume you discern it via crafting level limits given in DMG or XGTE, and assume the lowest level of crafter... So a +1 Weapon would be a valid target.
All this being acknowledged, I think a fair RAI would be to say the item in question is instead "disabled" until a Short or Long Rest, especially if it's "bonded...