Electronic – Raspberry Pi and USB modem – can I connect it like this

edit
I presumed the RPi could be USB powered, but Tony points out that it's not recommended. That seems to be an understatement. The RPI doesn't seem to have a specification (!), at least I couldn't find it, but I did find others who had the same problem. So in the end you have to rely on third-party sources. :-(

RPi models A and B consume 500 mA and 700 mA, resp. (Though even that doesn't seem to be a solid spec, as this page says "provisional, from alpha board".)

So, if by "external power supply" you mean a USB host or powered hub, then the answer seems to be no, since USB doesn't have to provide more than 100 mA without negotiating. You'll need another power supply, like a 5 V regulated wall-wart.

Many Wifi modules are designed to operate at 3.3 V, but if you can't find a 3.3 V regulated wall-wart you can use a 5 V type, and use an LDO (Low Drop-Out) post-regulator to go from the wall-wart's 5 V to 3.3 V.

Even if the module works at 5 V you may have a connection problem, since, as I understand it, the Raspberry works at 3.3 V. So the output level of the Raspberry may be too low for the WiFi module (or probably just high enough), and the input to the Raspberry may be too high. If the inputs aren't 5 V tolerant the high level may damage the device.

So 3.3 V is probably the way to go. The 3.3 V version of this power module can supply 750 mA.

If you prefer to get your power from a 5 V supply then the NX1117CE33 is a suitable part: it can provide more than enough current for the WiFi module, and accepts input voltages up to 20 V (so an unregulated 12 V input can be used as well). But if you use a high input voltage you'll have to keep an eye on power dissipation.

Seems reasonable at a glance, couple of thoughts:

• All the pins marked "GND" on the connectors will be connected together, so can be treated as a common ground. An easy way to test this kind of thing is by using a multimeter set to continuity mode (or ohms mode) and probe between the wires.

• I don't use Arduino, but I thought the DC in needed higher than 5V. According to this page, it needs between 7-12V. This makes sense if it's using a 5V linear regulator as it needs some headroom to regulate correctly. You may need to use the 12V rail, although lower is better (e.g. between 7V and 9V) if you plan on using a lot of current with the Arduino, as the on board regulator dissipates more power and gets hotter the higher the input voltage.
  EDIT - to clarify about input voltage:
  Looking at the schematic, the Arduino UNO uses a 5V NCP1117 linear regulator at it's DC jack input. From the datasheet we can see the dropout voltage is around 1V (hence the absolute minimum 6V input given in the link above). Supplying the input with 5V will produce an output of ~3.3V (1V + D1 diode drop) - the ATMega will probably still work at this voltage, but the 5V rail will no longer be a regulated 5V rail, so it's obviously not ideal.

• Yes, you can use the 12V rail to power the LEDs (assuming current limiting will be used correctly) The number will be limited by the current sourcing capabilities of the rail which should be in the supply specs. Up to 9 small RGB LEDS should be no problem though.