Do I need to keep all blocks when running Bitcoin Core? Do I need to keep all blocks, from block 0 up to the current one, when running Bitcoin Core, either as a full node or not? Or can I just keep the more recent blocks to save space?
Full nodes keep all blocks by default, but this is not necessary to achieve full node security. Full nodes validate the complete blockchain and enforce all consensus rules regardless of whether a full history is kept.
Keeping all blocks is a service to the network, as you'll be able to provide all blocks for synchronizing nodes or requests of thin clients.

If keeping the full blockchain is a problem, you can reduce the storage-footprint by running Bitcoin Core in pruning mode. Just add prune=&lt;n&gt; to your bitcoin.conf, where &lt;n&gt; is the amount of youngest blocks you want to keep in terms of mebibytes storage. The minimum is 550 MiB.