What happens when two nodes broadcast different next-blocks at almost the same time?

Nodes build on whichever block they saw first but keep the other branch; the tie breaks when the next proof-of-work extends one branch longer, and everyone switches to that longest chain.

Temporary forks are normal. A node works on the first valid block it received while saving the competing branch "in case it becomes longer." When the next block is found, one branch pulls ahead and nodes on the losing branch switch to the longer one.

The losing block becomes stale/orphaned; its transactions simply go back to being included later. This is why a brand-new block is the least safe — it might lose a tie — and why recipients wait for confirmations.