Arduino keeps crashing
It is inferred from your question that the Arduino is getting powered through the USB port.  As you say the Arduino appears "gone" from the computer when you light up too many LEDs.

All USB ports are only obliged to supply 500mA with out the USB Device (the Arduino) negotiating for more power from the USB Host.  If the load goes beyond 500mA, the USB Host (the computer) can limit the power my reducing the voltage.  At some point the Atmel processor on the Arduino will stop working.

There are many tutorial on how to use LED strips on the web.  Most will indicate power from a larger than normal power supply is necessary.  Also, that the LED strip power should be connected directly to an appropriately chosen power supply.

The following image from this web page illustrates this:



(Note: In this picture PIN 6 was chosen to communicate with the LED strip.  Your specific application (program or sketch) may use a different pin. Also, realize that the power supply in the picture is wired to only power the LED strip. The Arduino still needs to be powered.)