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My textbook says when heating potassium, rubidium, cesium with graphite, the metals react by invading the space between the layers. They form highly colored interstitial carbides that are non stoichiometric in nature (due to variable composition).
Do the intercalated metal atoms increase the conductivity ( maybe as alkali metal ions give away an extra valence electron) and density (because of interstitial defects) of graphite?
As you've already noted, heating the alkali metals on graphite causes the alkali metals to enter the interstitial spaces within the graphite layers.
An example of this would be $\ce{(CaC6)_n}$, whose structure is given below. The grey balls signify carbon and purple ones signify calcium
Do the intercalated metal atoms increase the conductivity ( maybe as alkali metal ions give away an extra valence electron) and density (because of interstitial defects) of graphite?
Let's talk about the more intuitive one first, density. The best way to understand this is by thinking about what exactly happens.
Graphite is a carbon allotrope that is made up of layers of sheets of carbon (graphene) where each layer has carbon atoms arranged in a trigonal planar manner. There are two types of graphite depending on how the graphene is stacked - α$-$graphene where the layers are placed in a hexagonal nature and β$-$graphene where the layers are placed in a rhombohedral nature but I digress.
Now, when an interstitial carbide is formed, the alkali metals are seen to occupy the spaces between the carbon atoms. Now, what happens if an atom replaces empty space? The mass increases volume remains the same and therefore density increases.
Now, time to talk about the conductivity. Here I refer to Carbon 1966, 4 (1), 125–127, which talks about the electrical conductivity of different types of potassium carbides.
TABLE $1$. RELATIVE CHANGES IN RESISTANCE WITH COMPOUND FORMATION
\begin{array}{lc} \text{Compound} & R/R_o \\ \hline \text{Initial Graphite} & 1.0\\ \ce{KC60} & 0.156\\ \ce{KC36} & 0.093\\ \ce{KC24} & 0.08\\ \ce{KC8} & 0.037\\ \hline \end{array}
Here $R$ refers to the resistance of the compound and $R_o$ refers to the resistance of the interstitial carbide. Therefore, as you can see the conductivity increases with the amount of alkali metal being added to the spaces between the layers.
