Will it burn.....

It depends on the connected component to the cap. The cap itself mostlikely will survive, cuz as you already said there is current limiting resistance (e.g. ESR). But a transistor or diode might get damaged.

Depends on the source mostly. The cap is made so it'll withstand current peaks, but the source may not be. For instance tube rectifier manufacturers always give a recommended limiting resistor before the first capacitor in their datasheet, and also a recommended capacitor value, which you should exceed at your own peril. But tubes are usually high impedance sources, and they have similar limitations. With transistors though, where the transformer can expect sudden peaks up to a few amps, it's usually not a problem when the initial surge is demanded. The transformer voltage will kneel to some degree in the beginning, and the internal resistance of the diodes will further limit the short circuit before the cap energises completely. The more important thing to consider is not whether the transformer will withstand the surge, but whether the diode bridge will. Usually when you choose a diode bridge you take one with a peak rated power at least 5 times the average one ( I usually take it with 10 times the capability). Other than that, I don't think you'll have a problem. Capacitors have internal resistance which will further dampen the surge.

Ah and about the rule of thumb you asked, it varies a lot, but for all intends of purposes I choose a capacitor with 1500uF for every amp of current. So for 5A of average current it'd choose 7.5mF, the closest value being 10mF (I usually round up to the next available value above the one I need)

And as bonus check this ;-) http://everycircuit.com/circuit/5236088859787264

Yes hurz is right with the bonus. Moreover capacitors have special places in the circuit, and I don't mean where they're just needed, I mean in the physical layout. Since they draw current in a very spikey manner, they can cause ground loop problems and interfere with the correct operation of some more sensitive parts of a circuit like a preamp for instance, where the noise is desirable to be kept very low. People usually try and eliminate these problems by using star shaped ground topologies, or increasing the ground surface area on the board.

hurz invented the word "niddle"

Shit, hope “niddle“ is not to offensive in your language ;-)