old floppy disks of different sizes. the bottom looks like 5 1/4" the ones on top with the metal centers are all 3 1/2". Both standards needed sleeves to be read. Many of these are likely trash now but that wouldn't stop me from trying to load them.
For 3.5", yes. 5.25" disks could be removed from their protective enclosures, inserted into a drive, and used as normal. At least until the exposed medium was damaged by fingerprints or other debris. Not something you would normally do though. Source: Did it myself a few times mostly out of curiosity.
This. When my favourite floppies started to have a worn sleeve (especially 3.5", where that metal protective covers started to bend out a bit, threatening to jam in the drive), I usually transplanted the disk itself over to a new sleeve.