Generally, powering up isn't as much of a problem, although certain peripheral equipment may not be identified during startup if they aren't already on when the computer tries to find them. Just put everything on a single shut-off switch and that problem should be solved. As for the shutdown process, certain operating systems leave files in strange "states" during operation and if the computer does not go through an orderly shut down the files will not be updated properly and will be corrupted next time your computer attempts to read them.
For removable media the above answer is true. If the hardware is a peripheral accessory device (scanner, printer, etc) the only risk is corrupting the operating system (typically & unfortunately Windows) drivers which could (but unlikely) have adverse and unpredictable results. The damage or dysfunction which could occur to the actual hardware are always described in the documentation which came with the device.
Hot-swapping refers to the removal and/or addition (swapping) of hardware into a computer system while it is running. On a normal desktop computer this is a very stupid thing to do and will result in damaged hardware and/or electrocution, but file servers are often designed specifically to allow the replacement of certain components (usually hard drives or power supplies) without powering the machine down. This is done by having redundant hardware that kicks in when needed allowing the device in question to be powered off and removed.
The two most important things to not do with a personal computer is to shut it off while the PC is powering down; and download information from suspicious websites. The two things a person should do with a PC is to back up the information on the PC on a regular basis; and to keep the PC dust free & in a place of room temperature.
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