I have sketch witch generate some sound using tone() function. I need louder sound so I plan to use LM386
http://www.national.com/ds/LM/LM386.pdf. I want to use first circuit from page 5
So questions are :
It will work with tone() function ?
Can I have separate voltage supply for arduino and amplifier (each powered from 9V battery ) ?
This circuit will handle 3W 4ohm speaker ?
It will work with tone() function ?
Yes. It will work with whatever you supply to its input.
Can I have separate voltage supply for arduino and amplifier (each powered from 9V battery ) ?
Yes, no problem.
This circuit will handle 3W 4ohm speaker ?
I think so, but word "handle" has to be cleared.
LM386N-1, LM386M-1, LM386MM-1 VS = 6V, RL = 8?, THD = 10% 250 325
LM386N-3 VS = 9V, RL = 8?, THD = 10% 500 700 mW
LM386N-4 VS = 16V, RL = 32?, THD = 10% 700 1000 mW
As you can see, LM386N-3 satisfy your requirements for Voltage, but not for R load and Power. Probably, it will works with 4 ohm load, the same time distortion level would be higher.
And power, of course, less than 25% of what speaker can accept.
I
Something else to keep in mind (and this is only anecdotal here) is that I have heard that the LM386 is very picky to develop with, on a breadboard especially. You can build your circuit exactly like the schematic in the datasheet, and it will fail to work. Best advice I have heard (but never tried) was to keep your leads and components as close to the LM386 on the breadboard as possible, keep the leads to the parts as short as possible, and don't take any shortcuts (and use the components specified in the reference design, unless you know -exactly- what you are doing). Apparently, a lot of these issues go away when using a proper PCB layout (but you still need to take into account routing of lines, length, etc). Likely it's a capacitance issue, from what I have gathered...
Distortion don't bother me too much because I need simple and moderately loud "beep" 
I plan to build it on universal board, it could solve some connection problems.
I noticed now that I bought N1 version...
Ok I see two NPN/PNP transistors connected together and connected to arduino pin using some potentiometer (10 kohm will be good ? ) Or one Darlington transistor
Abut second scheme , what capacitor I should use ?
Hmm this look way simple than using LM386.