I would lay out the cuts on the back side of the tile, heat tile with hair dryer or heat gun to soften it, spray cut lines with ammended water (soapy water) and make cuts with a razor knife. Best done outside and for piece of mind use a half face respirator with magenta colored filters.
To remove the old tiles, use the heat gun again to soften the one glued down then use a wide putty knife to carefully scrape them up. If the mastic is black, it could contain asbestos fibers too.
Don't know where you are located but under resilient floor covering institute (RFCI) guidelines and Texas laws a respirator is not needed for this response action, half face respirators and filer cartidges are cheap though. I would carefully wrap the old tiles, mastic materials, and putty knife it heavy plastic and properly dispose of the whole mess in a Type IV landfill.
I've been an asbestos inspector/management planner and O&M asbestos supervisor/contractor for many years, that's how I'd do it.