It looks like there are a few questions here, so I will try and split it up to what I think you mean.
how can someone access my site with different domain name?
Many domain registration services offer forwarding / redirection facilities, placing the website (say example.com) in an iframe. This would allow example2.com to appear as though it was example.com, very basic checks such as viewing the source code of example2.com would clearly show that it isn't the original website.
i.e. example.com displayed mysite.com actually... (at least, i want for local PC purposes).
Locally, you could set up DNS so that you type in a domain and it'll send you to the original website. If you change your DNS settings to your own DNS server, it will take precedence.
or how to modify REQUESTS using browser? are there any addons?
Unfortunately, I'm not to sure about such options for this.
If you could be a bit more specific about what you're after, I can try and narrow it down.