Is the gauge big enough? String Pixel
From link:

Quantity: 50 LEDs /string, 10 strings in total;
Operating voltage: DC 12V;
Power: 0.6W/LED

$$50 LED's \times 0.6W/LED = 30W / string$$
$$I = \frac {30W} {12V} = 2.5A/string$$
10 strings = 25A or 300W.  So 12V/350W power supply is fine.
30W / string assumes all LEDs are ON and each is drawing 0.6W.
So #22 AWG should be able to handle a single strand of 50 LEDs at 2.5A DC.  The actual capabilities means the shorter the wire to the power supply the better.  No extension cord between power supply and string.
But not connected in series.  All of the strings must be connected to the power supply separately or in parallel.  5A (two strings in series) would be too much for #22 AWG.

V+ to V+, V- to V-, data to data.  That's parallel.
The problem comes when you make statements like injecting power every 100.  That's series and that will fail.
Power 1, String 1, Power 2?, String 2, Power 3?.  No idea how that will even work.
You also have to factor in distance.  Longer from Power Supply, the greater the voltage drop to the wire.
So parallel.  All strings of LEDs leave the power supply, all V+ are connected together, all V-.
Data may be a problem.  If you connect all data, you may be controlling all LED 1 in every string.  All LEDs 2, etc.  So your data should probably be separate.
If you want to drive strings separately, separate 35W, 12V power supplys would allow each string to be powered.  65W, 12V drives two strings.