Several questions about the ADC LTC2323 function, I'm new in electronics
Wow, much questions.  I'll try some of them, however, the topics are far too vast to get complete answers in a few paragraphs.  You have some reading to do...

1-3) The chip is primarily designed to convert differential analog signals.  It can support 2 differential signal input pairs.  Each differential signal is comprised of 2 input wires.  The voltage on all of these input wires must always be greater than 0 volts (no negative voltages).  For a true differential input, the signal to be converted (or digitized), is Vadc = (Vin+ - Vin-).  Remember, both Vin+ and Vin- must be positive voltages, but if (Vin- > Vin+) then Vadc will be negatively signed.
There are other input configurations possible, but the general rules listed above will still need to be observed.
Refout is used to specify what the acceptable input voltage range is (see pg 16 of the datasheet).

4) Sample a voltage, and produce a number that when properly scaled, represents what voltage is observed in relation to the possible full scale range.  I this case, the full scale range for the produced number is 16 bits wide, resulting in a single bit representing 1 part in 65,535 possible unique values.  The actual result will not actually be this precise, but the chip will try its best.

For instance, if Vref = 4.096V, then the allowed voltage range for Vin+ = 0 to 4.096V, and for Vin- = 0 to 4.096V.  Vadc now has a range of +/-4.096.  This gives a best-case resolution of (8.192V/65,535), or 0.125mV.

The resulting Vadc is internally scaled for a range of 0 to 65,535 counts (the 16-bit range), with the digital value being represented as a signed integer (with a range of -32768 to +32768).  When Vadc = -Vref (-4.096V in our example), then the converted value would be -32768 counts.  When Vadc = 0V, then 0 counts.  When Vadc = 4.096V, then 32768 counts (and each voltage in between is digitized to its nearest corresponding count).

This number (with a possible value in the range of +/-32768) is sent out the serial line over 16 clock cycles.  One clock cycle per bit.

5) Yes, almost (but for 16 bits).  The conversion takes time, and there will be an amount of time uncertainty that you must understand.  See the datasheet, Understand the "Timing and Control" section (pg 20), and figure 21.  There are 2 serial output interfaces, one for each Vadc channel.

6) Understanding the timing is critical to getting the best performance from an adc.  Read and fully digest the datasheet.  You will need to understand every bit of it or you will not get the desired results from the chip.

The quality of signals that the external circuit supplies to the adc chip inputs is also critical.  Read about buffering &amp; driving adc inputs as well before committing a circuit design to a pcb.

Good luck.