That's kind of a weird footprint. Are those holes supposed to be plated? Even if you can get this to pass Altium's DRC your PCB fabricator may reject having copper right up to a hole like that without an annular ring.
Generally for weird pad shapes that can't be made using the standard pad objects you would use a region to create the copper areas, and then place a small pad somewhere inside that to create electrical connectivity to it (you'll need to set the mask and paste expansion properties of the region to ensure you can solder it). In this case, I would probably do the same, and make the region follow the shape of the hole (optionally with some clearance around it to make the PCB fab happy, if that will work with the connector). If you make the through holes unplated that may also prevent the short circuit error, but I'm not sure.
Assuming those pads and holes are just mechanical support, you can give them all the same designator, and then place one pin in the schematic symbol with that designator, so you can make one electrical connection in the schematic for all of the mechanical pins (since often they're all tied to the same metal shell in the part anyway). Even if you don't need an electrical connection to those pins, it's sometimes easier to connect them to gnd or whatever so you can stitch the pads to the ground plane with vias for better mechanical strength.
As to the errors, it looks like those pads aren't connected to a net (unless you have net name display turned off?). You can choose to allow short circuits for pads that are not in any net by adjusting your design rules, and that should eliminate this error--but could possibly allow other more problematic things to get through, so make these decisions carefully. There are a huge number of options when it comes to design rules, so it's hard to provide more specific advice with knowing how yours are set up.