Building a power supply, want to stay alive
Can 2.2 A of output current cause serious harm to my person or anyone else around?
Absolutely. From a non-shock perspective, it's enough amps to cause circuit failures to potentially smoulder and burn, or blow low-power devices to bits.
From a shock perspective, it's more than enough to be hazardous, since the mains voltage is high enough to force conduction through your body. On the secondary side of the transformer, not so much, thankfully.
Mains power is not to be toyed with lightly.
What happens if I short-circuit the secondary branch by accident?
The transformer will start drawing lots of current from the mains, causing:
Your appropriately-selected input fuse to open and safely isolate the circuit from the mains, or
Your inappropriate (or absent) fuse to not isolate the circuit, leading to your transformer sitting there and broiling, release some nice-smelling burnt varnish aroma (I call it "power supply incense") into the room. If it has a safety marking on it, it may sit like this indefinitely or fail in a "safe" manner (no dielectric breakdown, no flames or obvious smoke, etc.)
Is it a proper thing to do to use a 2.5 A fuse given that the laptop is 15 V / 1.8 A?
Depends on the load. Fuses not only have to be rated for steady-state current, but should handle things like inrush current (charging those capacitors), line surges and brown-outs without nuisance blows, as well as provide safety during abnormals.
Should I place it (a) between the transformer and the rest of the circuit or (b) between the circuit of the power supply and the laptop?
It must be on the mains side of the transformer. It must prevent any mains voltage from entering the circuit if it blows (which can get complicated if there's a protective earth involved, especially if people fuse the neutral and leave the line unfused). You should also consider a MOV (metal oxide varistor) between the fuse and the transformer across the mains to clamp any surges that come in (it will crowbar and open the fuse).
What does it mean when they say the laptop requires a rated power supply? Does it mean the power supply should employ a stabilizer?
Depends. I would interpret it as meaning regulated, and able to meet the voltage and current requirements of the laptop. I take it you're using the term 'stabilizer' to imply regulation, as that's not a common term in power conversion (at least to my ears).
Other points to ponder:
Keep a safe distance between the mains and the output. Creepage and clearance is important for life safety. Don't cross the barrier with any components unless they're safety-agency rated for such an application. Don't cross the barrier with any wires unless they're triple-insulated. Don't connect primary and secondary returns.
Consider adding over-voltage, over-current and over-temperature protection. Find a way to shut the thing down or blow the fuse if any of these bad things happen, to keep your downstream stuff from getting damaged by the power supply.
Can 2.2 A of output current cause serious harm to my person or anyone else around?
The current rating of the supply is not what determines if it is dangerous. What is dangerous is exposed voltages. If you contact the 240 V mains voltage, you will have an unpleasant experience.
In this circumstance, probably only a few hundred microamps or a few mA of current would be going through your body. But this is plenty enough to disrupt important biological processes like the beating of your heart.
Is it a proper thing to do to use a 2.5 A fuse given that the laptop is 15 V / 1.8 A?
It's a good idea to use a fuse, and your rating is reasonable. But this has nothing to do with the electrocution hazard. The fuse basically prevents your equipment starting a fire if there is a short circuit. It will do nothing at all to protect you if you touch the primary side of the transformer. First because fuses take too long to burn out to protect you from electocution; more importantly because any current shunted to ground through your body from the primary wouldn't be going through a fuse on the secondary side anyway.
Edit
It would be great to know that I can touch the secondary side without being hurt.
The key is to maintain proper isolation between the primary and secondary. Your transformer will do this. But also be sure to maintain a proper "creepage" distance between any wires on the primary side to any wires on the secondary side. Unfortunately I can't tell you the required separation off the top of my head.
I'm just Googling this mostly.
1) That's definitely not a healthy current, to say the least. Depending on the frequency, you can handle different amounts of current.link
2)It's a step down transformer(I think), so this link, says "If the secondary were shorted (the resistor replaced with a wire), the primary current would be much greater, and the secondary current would be 10 times the primary current."
3) I talked to a senior engineer at work, and he said to use a 2A fuse. I think you would put it between your power source and the transformer. That's what it looks like in this design. Scroll down a bit. Take a look at this.
4) I think those are called regulated power supplies.