The tiles look very much like "marley" ones if so they may contain asbestos.
Very common on solid floors I would go for laminate not anything self adhesive just lay the underlay with the inclusive DPM and the floor will be floating no need to tamper with the tiles more that you have to.
You will have some loose bits when you take up the gripper rod, clear up with a damp cloth not a hoover.
Plenty on line about Thermoplastic tiles, the black adhesive often contains asbestos too.
Don't panic though as bonded asbestos goes they are very low risk unless sanded.
If you want to seal any broken edges or remaining adhesive I used pva adhesive on mine.