You will not obtain a satisfactory result on this floor with any paint I am aware of. Particularly with damaged tiles, the tile and repairs will telegraph throught the paint. The very hard tile is most likely an asphalt tile and may or may not contain asbestos. It can be removed safely, but Do not sand.
I suggest you level the floor with floor leveling compound. You will usually find this in the tile section of a store where grout is sold. Once the floor is level, you can use self-adhesive vinyl tiles or maybe a new carpet or sheet vinyl remnant to make an inexpensive finished floor. It seems unreasonable for the landlord not to participate at all in this, perhaps you could negotiate a split of material costs and you providing installation.