I think Q1 is false and a "simple" simulation can confirm (see below). However, there is probably an equation that can relate $\rho(t_1, t_2)$ to $\rho(x_1, x_2)$. For Q2, I think, it is basically related to the residual error in $y$.
The way I see it is basically to take the "complete" linear model: $y=\beta_1x_1+\beta_2x_2+\epsilon _y$,or in matrix algebra : $y=BX + \epsilon_y$. Here, the $\beta$ in $B$ are partial regression coefficients, thus they are not the same as in the OP (where they are simple regression coefficients).
Let's define $\Sigma$, the covariance among $X$, then we know: $B$, $\text{var}(B)$ and $(1-R^2)$, which is the residual error variance in $y$.
Regarding OP, we can already compute
$[\beta_1^{*},\beta_2^{*}] =B^{*}= B\Sigma$.
Since we are regarding the linear model as two different models, there is then two $R^2$ values, one for each linear model, which are $R^{2*}=(B\Sigma)^2$.
We now see that $\Sigma$ becomes $\Sigma^*=\text{diag}(\Sigma)$, which is a square matrix with variances in the diagonal and 0 in the off-diagonal elements.
On the $t$ values, in the complete linear model we get :
$t=\frac{B}{(\frac{(1-R^2)}{ \text{diag}(\Sigma^{-1})})}$ and $t^*=\frac{B^*}{(\frac{(1-R^{2*})}{ \text{diag}(\Sigma^{*-1})})}$. So the inverse of $\Sigma$ and $\Sigma^*$, the regression coefficients, and the $R^2$ are very different. We see where the similarities come up, but know the complete relation.
For the simulation :
n = 10000
b1 = .4
rho = .3
b2 = .2
reps = 1000000
tvalue = matrix(0,4,reps)
for(i in 1:reps){
X = MASS::mvrnorm(n=n,mu=mu,Sigma=S)
Y = t(B%*%t(X) + rnorm(n) * (sqrt(1-R2)))
res0 = lm(Y~X[,1]+X[,2])
res1 = lm(Y~X[,1])
res2 = lm(Y~X[,2])
tvalue[,i] = c(summary(res0)$coefficients[2:3,3],
summary(res1)$coefficients[2,3],
summary(res2)$coefficients[2,3])
}
> round(cov(t(tvalue)),3)
[,1] [,2] [,3] [,4]
[1,] 1.198 -0.242 1.155 -0.068
[2,] -0.242 1.020 -0.062 0.917
[3,] 1.155 -0.062 1.273 0.268
[4,] -0.068 0.917 0.268 1.080
> round(cor(t(tvalue)),3)
[,1] [,2] [,3] [,4]
[1,] 1.000 -0.219 0.935 -0.060
[2,] -0.219 1.000 -0.055 0.874
[3,] 0.935 -0.055 1.000 0.229
[4,] -0.060 0.874 0.229 1.000
We see that they are close (.229 is close to .3), but not the same.