  A detailed exact-diagonalization study is made for the doping dependence of
the single-particle spectral function $A({\bf k},\omega)$ and momentum
distribution function $n({\bf k})$ of the two-dimensional $t$$-$$J$ model as a
representative model for doped Mott insulators. The results for $A({\bf
k},\omega)$ show unambiguously that the rigid-band behavior is realized in the
small-cluster $t$$-$$J$ model: upon doping, the uppermost states of the
quasiparticle band observed at half filling simply cross the Fermi level and
reappear as the lowermost states of the inverse photoemission spectrum, while
the photoemission side of the band remains essentially unaffected. We discuss
problems in directly determining the Fermi surface from $n({\bf k})$ and make a
situation where they are largely avoided; we then find clear signatures of a
Fermi surface which takes the form of small hole pockets. The identical scaling
with $t/J$ of the quasiparticle weight $Z_h$ and difference in $n({\bf k})$
between neighboring ${\bf k}$-points suggests the existence of such a Fermi
surface in the physical regime of parameters. We construct spin-bag operators
which describe the holes dressed by the antiferromagnetic spin fluctuations and
find that elementary electronic excitations of the system can be described well
in terms of weakly-interacting spin-1/2 Fermionic quasiparticles corresponding
to the doped holes. We make a comparison with other numerical calculations and
recent angle-resolved photoemission experiment and argue that, adopting this
rather conventional Fermi-liquid scenario with non-Luttinger Fermi surface, one
would explain many quasiparticle-relating properties of doped cuprates in a
very simple and natural way. We also show that the dynamical spin and charge
excitations deviate from the particle-hole excitations of this Fermi liquid:
e.g., the dominant low-energy spin excitation at momentum transfer $(\pi,\pi)$
reflects excitations of the incoherent spin background and is identified as a
collective mode comparable to spin waves in the Heisenberg antiferromagnet.
