Atmega328 output pins are rated at 40 mA absolute-maximum current, and the 328's total current through Vcc and ground pins is 200 mA absolute max. Eg, see table 29.1 in the Atmega328 specs.
Although some sources rate a high-speed 716 coreless dc motor at 40 mA, other sources rate it at 100 mA. In either case, it exceeds normal I/O pin capability. You should use FETs or other drivers between Arduino output pins and the motors rather than trying to drive the motors directly from digital output pins.
Edit: Regarding the comment, “But most projects i've seen don't use any drivers, Why so ?” I replied:
Perhaps they are using slightly higher voltage to the Pro Mini, eg 3.7 V up to 5 V, which after accounting for voltage drop in the i/o pin circuits may deliver enough voltage to the motors. Note that a driverless circuit ie direct drive from the i/o pins operates the 328 out of its safe operating area (SOA), but it may be close enough to the safe area to usually work.
For example, some of the 716-based quadcopter pages I looked at say they used 4.2 V LiPo batteries. Also note, 716 seems to be a fairly generic motor designation; it refers to the motor's 7 mm diameter and 16 mm length. As a generic designation it has been used for several different motors, some of which produce less power. For example, in thread #1710948 on rcgroups.com, people refer to different RPMs, less lift, etc. for various motors, and the importance of shaving grams off of the copter's weight to allow it to fly.
Another item that matters when drawing high current from an Atmega328 is which ports are used; and for some devices, it matters whether drive is high-side or low-side, ie whether current is sourced or sunk. The Atmega328 spec sheet (Atmel-8271-8-bit-AVR-Microcontroller-ATmega48A-48PA-88A-88PA-168A-168PA-328-328P_datasheet_Complete.pdf) says “Each output buffer has symmetrical drive characteristics with both high sink and source capability”, but for several kinds of drivers less voltage is lost with low-side drive than with high-side. Ie, you could try connecting the motor's V+ lead to Vcc and its V- lead to the i/o pin, and inverting the on-off logic in the sketch. (I don't know if this will make any difference at all in how well things work.)
Regarding which ports are used, the footnotes to Table 30-1, “Common DC characteristics”, spell out various combinations of ports where total current draw must not exceed a limit (100 mA in some cases, 150 mA in others). On the Uno and Pro-Mini, digital outputs marked 10 through 13 (as used for motor control in the question's sketch) are port B Atmega pins, with a footnoted total-current limit for the group. If you move two of the motor drives to outputs marked 2...4 or D2...D4, then a different total-current limit will apply. (Again, I don't know if this will make any difference at all in how well things work.) Refer to Table 30-1 footnotes for details.
As noted in Gerben's comment, having flyback-diodes for the motor circuits is a good idea. This consists of placing power diodes across the motors, cathode to V+, anode to V-, to prevent energy stored in the motor's inductance from producing extreme i/o pin voltages each time the motor is turned off.
Edit 2: Regarding “which diode, FET to use and where [to] learn about them”, electronics-tutorials.ws is an accurate and helpful presentation about FET digital circuits. In particular, see the section called “An example of using the MOSFET as a switch”.
Which FETs to consider depends in part on your electronic-circuits fabrication skills. Note that surface mount devices (SMD parts) usually weigh less than other versions of parts, which is important for a lightweight quadcopter. However, some people find SMD parts difficult to work with. In my opinion, the main difficulty is holding parts in their proper places while soldering them, due to small size. (Eg, a SOT-23 SMD transistor package is on the order of 3mm x 2mm x 1mm, and a diode may be half as big.) If you have good soldering skills and a way to hold parts in place while soldering, you should be ok, either using a printed circuit board or air-wiring parts using wire-wrap wire (ie, 30 AWG wire, about 0.25mm diameter).
Some typical logic-level-input SMD FETs are shown in mouser.com's online catalog. Note that in that list, RDS_on decreases a bunch as unit price increases slightly. For example, the FDN359BN would have under 0.1 ohm resistance when operated with Vcc = 3.7 V and currents < 100 mA. Voltage drop running a 40 mA motor should be about 4 millivolts.
Some typical SMD Schottky diodes also are shown in mouser.com's online catalog. I think even the cheapest of these would work ok. Of course you'll need to adapt all these suggestions to your own skills, sources of parts supply, and budget.