Quick questions on Minimum spanning trees and Prim's algorithm for HSC Maths Standard 2

A spanning tree of a connected graph is a subgraph that:

A minimum spanning tree is a spanning tree with the smallest total edge weight among all possible spanning trees of the graph.

The diagram above shows the finished minimum spanning tree. Here is the same graph with Prim's algorithm run one edge at a time. This lets you watch the tree grow and follow the reasoning at each step. Tree edges are heavy and in colour, and vertices already in the tree are ringed.

Kruskal's algorithm sorts all edges and adds them in order, skipping any that create a cycle:

An edge creates a cycle if both of its endpoints are already linked, either directly or through edges you have already chosen. For small graphs you can usually see this by eye. For larger graphs, keep track of which vertices sit in which connected piece (called a component) as you add edges.

The cheapest unused edge sometimes creates a cycle. Skip those.

Sum all the chosen edge weights at the end.

A spanning tree must connect every vertex. If your $n - 1$ edges form a forest (disconnected components), you have made a mistake.