  The first "seeds" of supermassive black holes (BH) can range from
$\sim10^2-10^6~M_{\odot}$. However, the lowest mass seeds ($\lesssim10^3
M_{\odot}$) are inaccessible to most cosmological simulations due to resolution
limitations. We present our new BRAHMA suite of cosmological simulations that
uses a novel flexible seeding approach to represent low mass seeds. Our suite
consists of two types of boxes that model $\sim10^3~M_{\odot}$ seeds using two
distinct but mutually consistent seeding prescriptions at different simulation
resolutions. First, we have the highest resolution $[9~\mathrm{Mpc}]^3$
(BRAHMA-9-D3) boxes that directly resolve $\sim10^3~M_{\odot}$ seeds and place
them within halos with dense and metal poor gas. Second, we have
lower-resolution and larger-volume $[18~\mathrm{Mpc}]^3$ (BRAHMA-18-E4) and
$\sim[36~\mathrm{Mpc}]^3$ (BRAHMA-36-E5) boxes that seed their smallest
resolvable $\sim10^4~\&~10^5~\mathrm{M_{\odot}}$ BH descendants using new
stochastic seeding prescriptions calibrated using the BRAHMA-9-D3 results. The
three boxes together probe BHs between $\sim10^3-10^7 M_{\odot}$ at $z>7$ and
we predict their key observables. The variation in the AGN luminosity functions
is small (factors of $\sim2-3$) at the anticipated detection limits of
potential future X-ray facilities ($\sim10^{43} \mathrm{ergs~s^{-1}}$ at
$z\sim7$). Our simulations predict BHs $\sim10-100$ times heavier than
expectations from local $M_*$ vs $M_{bh}$ relations, consistent with several
JWST-detected AGN. For different seed models, our simulations merge BH binaries
at $\sim1-15~\mathrm{kpc}$, with rates of $\sim200-2000$ per year for
$\gtrsim10^3 M_{\odot}$ BHs, $\sim6-60$ per year for $\gtrsim10^4~M_{\odot}$
BHs, and up to $\sim10$ per year amongst $\gtrsim10^5 M_{\odot}$ BHs. These
results suggest that the LISA mission has promising prospects for constraining
seed models.
