Deleting edges such that largest connected component has at most $n/4$ nodes
One naive thought would be that we can maintain the list of the size of current connected component. After removing an edge the content of the list will be changed and when the max member of this list decreased to no more than $n/4$ we can terminate and output the edge. However, simulating such process seems inefficient.

Though removing edge is somehow hard for us to maintain the size of connected component, adding edge is not.

We can consider the problem reversely:

Initially, there is no edge in graph $G$, then we add edge in the order of $e_n,e_{n-1},\cdots,e_1$ and at the end of this process $G$ is connected. Therefore there an edge $e_i$ such that just before adding $e_i$, every connected component of the graph has at most $n/4$ nodes, but after adding $e_i$, the graph has at least one connected component with more than $n/4$ nodes.

Obviously, such $e_i$ is just the edge we want for the origin problem.

We can use Union-Find Algorithm to maintain the size of each connected component. When adding an edge$(u,v)$, first check whether $u,v$ are in different connected components. If yes, then we get a bigger connected component and the size of it will be the sum of size of those two components, otherwise we do not need to do anything.

Once we find adding $e_i$ results in a connected component bigger than $n/4$, we can terminate and output it.

The overall time complexity is $  O(m\alpha (n)) $