This question is a duplicate, but the very simple answer is yes, absolutely.
Every atom has a gravitational force with every other atom.
Yes, the Cavendish device can quite easily measure gravity between two everyday objects, as long as they weigh about a KG or so ... It's that simple.
For even smaller objects, it might be difficult to measure - but so what? (Any number of physical qualities are difficult to measure, when very small or very large.)
A simple way to think about it - consider the Earth. It obviously has gravity right? Now consider one small piece of the earth (just some random piece of rock inside the Earth). All those little pieces ... have to have gravity right? Or else the whole thing overall, would not have gravity. That thought experiment can help you see that "even your pencil has gravity".
What is the current scientific consensus on this?
100.0% of all scientists believe all atoms in the universe attract all other atoms gravitationally.
Do small objects (like my pencil) exert forces on other small objects (like my eraser),
(Note that -- very simply -- your pencil is attracting every tiny little rock that makes up the Earth. Consider the Earth as being trillions of pencil-sized little rocks. So, that's gravity.)
or is it only large celestial objects that can exert a gravitational force?
No, it very much applies to every single atom.