Selecting connected subgraph that exceeds value c, with least possible weight
This problem is NP-hard.
Let $S = \{x_1, \dots, x_n\}$ be an instance of partition.
Create a clique $G$ on $n$ nodes $v_1, \dots, v_n$.
Set both the cost and the weight of $v_i$ to $x_i$.
Set $t = \frac{1}{2}\sum_{x_i \in S} x_i$.
If there is a subset $C$ of $S$ such that $2 \sum_{x_i \in C} x_i = \sum_{x_i \in S} x_i$, then the set of vertices $\{v_i \mid x_i \in C\}$ is connected, has total value $t$ and total weight $t$.
If there is no subset $C$ of $S$ such that $2 \sum_{x_i \in C} x_i = \sum_{x_i \in S} x_i$ then every subset of vertices of $G$ either has total value smaller than $t$ (and hence is not a feasible solution), or has a total value larger than $t$, and hence also weight larger than $t$.
Then you have that the answer to the instance of partition is yes if and only if the optimal solution to your problem has measure (total weight) $t$.