The future Electron-Ion Collider (EIC) will operate a series of high-luminosity high-energy electron+proton ($e+p$) and electron+nucleus ($\textit{e + A}$) collisions to study several fundamental questions in the high energy and nuclear physics field. Heavy flavor hadron and jet production at the EIC plays an important role in exploring both potential modification on the initial-state nuclear parton distribution functions (nPDFs) and final-state parton propagation and hadronization processes under different nuclear medium conditions. The current design of the EIC ePIC detector has good performance of vertex and track reconstruction, particle identification and energy determination in the pseudorapidity region of $-3.5<\eta<3.5$, which will enable a series of high precision heavy flavor hadron and jet measurements. Latest simulation studies of the projected nuclear modification factor $R_{eA}$ of heavy flavor jets and heavy flavor hadron inside jets in $e+p$ and $\textit{e + Au}$ collisions at $\sqrt{s} =$ 28.6 GeV and 63.2~GeV as well as the projected statistical accuracy of inclusive and differential charm baryon over meson ratio measurements in $e+p$ collisions will be presented. The impacts of these proposed EIC measurements on constraining the heavy quark propagation properties in cold nuclear medium and exploring the heavy quark hadronization process will be discussed.