First up, the RAW. This is pretty simple: Divine Sense works on celestials, fiends, and undead. Player characters are all humanoids (see page 11 of the PHB), and the Tiefling traits do not say anything about making the character a fiend in any way.
Next, we have the lore. In 3.5e, tieflings were the descendants of actual fiends, whereas in 5e, they are simply a race that has had the essence of Asmodeus infused into their bloodline. So they're a bit less fiendish than they used to be. If we look at the fiend type in the Monster Manual, it says that
Fiends are creatures of wickedness native to the Lower Planes.
Tieflings definitely don't fall under this category lore-wise, because they are a humanoid race native to the Material Plane.
Trying to argue that abilities that work on fiends should work on tieflings is like trying to argue that abilities that work on dragons should work on dragonborn, but what you have to remember is that fiends are the physical embodiment of evil. The differences between tieflings and real fiends are insurmountably greater than the differences between dragonborn and dragons.
The taint in the tiefling bloodline is a devilish one, so if they were fiends, they would be devils. Taking a look at the description of devils in the Monster Manual, we see that
Devils personify tyranny,
and that
Devils live to conquer, enslave, and oppress.
The fact that tieflings are allowed to be any alignment contradicts this, and similarly, tiefling characters are allowed to do things other than conquer, enslave and oppress - they're definitely not devils.
Finally, if another argument is needed, Jeremy Crawford agrees that tieflings are humanoids, not fiends, and can't be sensed by a Paladin's Divine Sense.
No, Divine Sense won't allow you to hit it as if it was visible. Invisibility says that:
Attack rolls against the creature have disadvantage,
and the creature’s attack rolls have advantage.
and Divine Sense does nothing to change that. So what's the advantage of sensing them? Well, the advantage is that you sense them. If you use Divine Sense and discover that you're surrounded by undead and fiends, isn't that better than not knowing?
You could make the argument that if you knew to use Divine Sense, there wouldn't be any point to using it. This isn't really true, though: there's a big difference between knowing that there's something lurking nearby and knowing how many, where, and approximately what creatures are lurking nearby.
Finally, the most important benefit: knowing where it is means that you know where it is. This allows you to use spells like Dispel Magic to remove its invisibility, or just spells like Fireball to hurt it and invisibility be damned. For that matter, you said that you still have disadvantage hitting it, which is true, but at least you can try to hit it. (If you didn't have Divine Sense, you wouldn't even know where it is, you'd just have to swing your sword at a random square and hope. Unless you hit, you won't even know if you guessed right.)
Best Answer
No, RAW it cannot detect that.
The text of the paladin's Divine Sense feature reads:
It continues:
So there is nothing about sensing cursed objects. Moreover, let's look at an actual cursed object in the DMG. For example, the armor of vulnerability states explicitly:
This doesn't mean there cannot be an ability or item that would recognize a curse in such an object, but the paladin's Divine Sense is not it.