I installed the SmoothStepper V1.2 board into my system and I found out that the home switches don't work. They worked before installation of the board. How can I fix this problem? How do I start Mach3 software with the SmoothStepper profile instead of the Mach3Mill profile?

Easiest way is to just go into the properties of your Mach3Mill shortcut and change the profile info in its target box. Here is an example
C:\Mach3\Mach3.exe /p Mach3Mill (this is the standard profile)
C:\Mach3\Mach3.exe /p SmoothStepper  (this is your SS profile)
Whatever your profile is called you want that typed after the /p, notice however that there is a space between the C:\Mach3\Mach3.exe  and  /p SmoothStepper

Home switches should work fine, sounds like you may not have them configured properly, open your old profile and have a look taking notes of what you have them set as then shut Mach down and open your SS profile and see if they are different.

Hood

I got three resistors in the package but I don't know what they are for.

The info for them  is in the manual on the SS site, they are for termination of the differential inputs if needed.
Hood

When the SmoothStepper board is connected to my computer and I try to turn my computer on, nothing happens until I turn my computer off and on. The first time I try to turn my computer on, the power LED on my monitor is still yellow instead of green. When I turn my computer off and on, my computer boots normally. If I turn my computer off, disconnect the SmoothStepper board from my computer and turn my computer on, my computer boots normally. I tried to cut out a part for a model of a subway car through SmoothStepper but something went wrong and the part was ruined. Is something wrong with the SmoothStepper board or I didn't set up Mach3 program properly?

What kind of computer do you have? Sounds like it may be having issues with the USB, maybe trying to power the SS externally would help, make sure you use a regulated 5V supply and remember to shiift the jumper over on the SS for external power.
Hood

I have a desktop computer with Intel D865PERL motherboard. I'm not sure if the 5V power supply that I bought for my Probotix PBX-RF breakout board is regulated or not. I don't have access to my CNC router from now until Monday morning because I'm at home and my CNC router is in the wood working shop at the program for mentally handicapped people where I work.

Motherboard should be capable enough, maybe have a look in the BIOS and see if Wake on USB is enabled. Dont really see how it would be a problem but you never know and is definitely worth a look .

Hood

What about self-powered USB hubs?

Self powered hubs should work fine with the SS.
Hood