Merhaba.

I have always been convinced that the word osmanlı and Ottoman sultans names starting with Osman have the same origin and that is from Arabic language and from the Muslim cultural field in general. I don´t know any Arabic word which sounds like this, but perhaps my idea came from the fact that the third of the Rashidun caliphs, Prophet´s companion- Uthman is also sometimes referred to as "Osman" when his name is oficially written in Arabic alphabet as عثمان - so is the name of the first sultan of the Ottoman Empire - Osman I (Gazi).

Surprisingly, today in a university class of Turkish early history I was told that the "Osmanlı İmparatorluğı" was named after one of the Turkish tribes originating from the Seljuq Turks. So finally, what is the connection between the word "Osman(lı )" in Turkish history and the caliph Osman who didn´t have much to do with Turks after all? Did one of them take the name after another or is it just a conincidence?

ottoman emperors line comes from osman bey, who comes from kayı boyu, which can be traced back to middle asia. if you look at the personal clothing and armor of each one of the turkish emperors and rulers you will notice the asian pattern, helmets which look like samurai or mongol helmets, later combined into the ottoman pattern, sarıks and robes and fes and later uniforms.

There is no such a tribe whose name is Osman or Osmanlı. Ottoman Empire was named after the founder and the first sultan Osman as you say. It is known that Ottoman family is a member of "Karakeçeli" branch of "Kayı" tribe. 

http://en.wikipedia.org/wiki/Kay%C4%B1_tribe

So, finally the name Osman came from the third caliph name- Uthman?

Yep. But it is not a name of a tribe.

It is just like ´George Washington´s name George is originally from ancient Greek but there is nothing between being Greek or Latin or whatever and Washington or being American / from USA. That is just a name.

Osman was the name of the first sultan of this dynasty. There is no real relation between him and 3rd caliph.