Why is spacetime curved by mass but not charge?
Charge does curve spacetime. The metric for a charged black hole is different to an uncharged black hole. Charged (non-spinning) black holes are described by the Reissner–Nordström metric. This has some fascinating features, including acting as a portal to other universes, though sadly these are unlikely to be physically relevant. There is some discussion of this in the answers to the question Do objects have energy because of their charge?, though it isn't a duplicate. Anything that appears in the stress-energy tensor will curve spacetime.
Spin also has an effect, though I have to confess I'm out of my comfort zone here. To take spin into account we have to extend GR to Einstein-Cartan theory. However on the large scale the net spin is effectively zero, and we wouldn't expect spin to have any significant effect until we get down to quantum length scales.
Gravitation couples to anything within the stress-energy tensor, as dictated by the field equations,
$$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R = 8\pi G T_{\mu\nu}$$
Charge and angular momentum both affect the curvature of spacetime, as they affect the metric. For example, consider a spinning charged black hole, desribed by the Kerr-Newman metric,
$$\mathrm{d}s^2 = -\left( {\mathrm{d}r^2 \over \Delta + \mathrm{d}\theta^2} \right)\rho^2 + \left(\mathrm{d}t-\alpha \sin^2 \theta \mathrm{d}\phi\right)^2 \frac{\Delta}{\rho^2}-\left( (r^2+\alpha^2)\mathrm{d}\phi -\alpha \mathrm{d}t\right)^2 \frac{\sin^2 \theta}{\rho^2}$$
The parameters $\alpha$ and $\rho$ depend on the angular momentum, and $\Delta$ in fact does depend on the charge of the black hole. Evidently, the curvature forms are also dependent on these.
Why isn't spacetime curved due to other forces or aspects of bodies?
Spacetime is affected by the presence of other fields. For example, electric and magnetic fields are described by the Maxwell Lagrangian,
$$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$$
where $F=\mathrm{d}A$ is the field-strength, a closed $2$-form. The field theory has a non-vanishing stress-energy tensor (derived by applying Noether's theorem to spacetime translations) which sits on the right-hand side of the field equations, and will induce curvature. Another example: the Kaluza-Klein metric in Kaluza-Klein theory is given by,
$$\mathrm{d}s^2 = g_{\mu\nu}\mathrm{d}x^\mu \mathrm{d}x^\nu -e^{2\sigma(x)}\left[ \mathrm{d}\psi + A_\mu \mathrm{d}x^\mu\right]^2$$ Hence, in this $5D$ model spacetime is influenced by a scalar field $\sigma(x)$, and a four-potential $A_\mu$.
For completeness, the action which the Kaluza-Klein metric gives rise to is,
$$S=-\frac{1}{16\pi G}\int \mathrm{d}^4 x \, \sqrt{g_4} \, \mathrm{d}\psi \,e^{\sigma} \left[ R^{(4)} + \frac{1}{4}e^{2\sigma}F_{\mu\nu}F^{\mu\nu} -2e^{-\sigma} \square e^{\sigma}\right]$$
which reduces to Einstein-Maxwell theory if $\psi \sim \psi + L$, for some period $L$ and the dilaton $\sigma=\mathrm{const}$.
The unique property why mass and no charge respectively spin (at least not strongly, only by "side-effects") curves space-time is the equivalence principle. The equivalence principle says that gravity mass == inertial mass. After thinking hard about this property (experimentally proven by Galilei) Einstein found out that as a consequence space-time must be curved by mass.
On the other hand there no compulsory relationship between the charge (or spin) and the inertial mass, better said, there is no relation at all. Therefore charge or spin have a priori no effect on space-time, at least not a direct one. As other said, the electromagnetic field carries energy and via its energy it contributes to the curvature of space-time. But charge is not the source of space-time curvature, this is reserved to mass respectively energy due to the equivalence principle