That is a great question. You can buy safety strips to apply to the bottom of the tub and I think that this would be your best bet. Hotels will buy the tub with a safety bottom already etched in the surface. You can get an etching compound to do the bottom of the tub but you will find the surface will be tough to keep clean.
Yes, by removing your tub and installing a wheel chair accessible shower base complete with a custom-formed one-piece acrylic wall system. As for shower doors you may need to go with a curtain, because of the bottom wedge or track for the doors.
Rock salt will work but it will cause the most damage to the wood. You will be better off trying to shovel and remove any snow and ice and then use a mixture of sand and potassium chloride or calcium chloride, both commonly called snow melt pellets. These work at a lower temperature and will be less damaging to the wood and surrounding vegetation.
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If it was me and my parents were living there I would personally go with the tile, its warmer, less slippery and easier to keep clean. You might consider mixing it up for effect but overall carpet for now.
I'm not aware of what type of fire alarm you mean, a house unit should be checked twice a year and the battery replaced. In a condo or apartment building there may be condo rules that specify how often they are checked, or there should be.
I would want to install the gab bars directly into the underlying wall studs. First you'll need to locate the studs (using a "stud finder" available from your home store). Then check out how and where you want to position the bars, and make sure they will line up with the studs (you may have to put them on at an angle to align with the studs). Mark the position of the mounting brackets for the grab bars and then drill a small pilot hole through your tiles (using a 1/8" carbide drill bit). Now, drill a larger second hole to accommodate the mounting screws using a carbide drill bit, then install the mounting bracket and grab bars. Finish by applying some bathtub caulk around the top of the mounting brackets to prevent any water getting in under your tiles. Make sure to use bars specifically designed to be used as grab bars and strong enough to support a person's weight. Towel bars aren't strong enough to do the job.