Distributions on the sphere is studied and used in directional statistics, see for instance the text Directional Statistics by Kanti Mardia & Peter Jupp. The entropy of the uniform distribution on the $n$-sphere $\mathbb S^n \subset \mathbb R^{n+1}$ is by my calculations $\log\{\frac{2\cdot \pi^{(n+1)/2}}{\Gamma((n+1)/2)}\}$, which simply uses the surface area of the sphere.
For the non-uniform distribution given, by the sampling process given the radius squared $R^2=\| x\|^2$ will have a non-central chisquared distribution woth non-centrality parameter $\lambda=t^2$. Then by calculating the conditional density given $R^2=1$ we can find the density on the sphere (details not given) as proportional to $\exp(t x_1)$, which by results in chapter 9 of the Mardia&Jupp book proves this is a von Mises-Fisher distribution. By the same reference, there is a maximum entropy characterization of the von Mises-Fisher distribution with given expectation, see also. With this results it shouldn't be difficult to calculate the entropy directly. And, indeed it is calculated here.