Can host chane my design?

Hello there, i'm new here! Nice to meet you all.
So, i have one little, but big problem, well.
Can my hosting change my design? I'll give you some screenshots who can give you more information of my words here. :tickle:
My last work on my localhost looks like : Screenshot by Lightshot (Here is button for fast going on top of page. It's right bottom.)
When i put this my work on my online web server then looks like : Screenshot by Lightshot (Buttons is now wrong, wrong part of page, i don't design like that. Now is left bottom...)

Ps. Interesting is that, when i put this my work on my EasyPhP problem is similar.

Second. 'What to listen?'
On localhost my work looks like this : Screenshot by Lightshot it's not problem, but what to listen, i used Bootstrap classes for make button my navigation on xs screens, they looks like this : Screenshot by Lightshot When i fiscally change screen size of my web browser, but when i used Google Chrome size view project looks like this: Screenshot by Lightshot
What to listen? Fiscally change screen or Google Chrome screen view? How will be look on phone?

Thank you.
Have a nice day.