Full nodes do not always relay, or always receive transactions if they have this setting turned off. The operator of a node can use the "blocks only" setting to only receive new blocks, and is a way to reduce bandwidth. There can also be instances when a miner will receive transactions to consider including in their blocks from not-full nodes, such as their websites, or from SPV clients (light nodes) if the miner's node is an electrum server for example.

A full node is one that validates each block in full, including that transactions included in said block contain valid signatures and have not previously been spent (are not double spends).

The decision between running a full node and a SPV client (a "light node") depends on the end user's value of security and privacy. A full node will use more resources, such as bandwidth, computing power, memory and hard drive space, and a SPV client will have incrementally lower security and privacy, but I am not aware of any thefts of coin that were stolen/lost because the end-user was using a SPV client.