3A lamp on 5A outlet

You should be OK to use a 3A appliance on a 5A circuit. The circuit is capable of supplying up to 5A but it is the individual appliance (lamp in this case) that determines how much current is drawn.

Round-pin plugs vs rectangular-pin plugs

Modern usage

I'm surprised a new home would have BS 546 round-pin plugs instead of the more usual BS 1363 rectangular-pin plugs.

Wikipedia says

Although BS 546 plugs and sockets are still permitted in the U.K. and are used for special purposes, new installations since 1947 generally have used BS 1363 sockets and fused plugs.

Until now, I thought the only place where round-pin plugs were used was in theatrical stages, where I think they are used to prevent people plugging inappropriate devices into theatrical lighting circuits.

Safety

If the BS 546 plugs don't have built-in fuses, they don't provide any protection to the line-cord between the plug and the lamp.

BS 1363 plugs have additional safety features which may not be present in BS 546 plugs

• fuse to prevent overload of appliance cord (& hence fire.)
• shutter in socket opened by earth pin (safer sockets).
• insulating sleeves around base of live and neutral pins.
• high insertion force to reduce risk of partial removal.
• designed so earth wire is last to break under cord strain.

For example a small child can poke their hands (or part of a metal toy) behind a loose BS 546 plug and be in contact with 230 V.

According to the article at this link, replacing an ungrounded receptacle with an ungrounded GFCI receptacle will offer more safety than an ungrounded receptacle (with presumably a 2-3 prong adapter). A fault in the plugged-in appliance will still trip the GFCI even though there is no path to ground. This seems highly superior to using a 2-3 prong converter with a do-it-yourself ground path, as that would just allow any stray current to follow the ground path versus tripping a GFCI.

http://communities.leviton.com/thread/1080

I don't know why a fridge would trip a GFCI.

Assessing your risk is a bit tougher. In my simple mind, user-triggered electrical faults come in two basic varieties - overloading the circuit or providing an alternate current path to ground. Do you plan to run extension cords under carpets? Are your electrical items in good condition - e.g. relatively new, no frayed cords, etc? Do you plan to use any appliances in wet conditions? GFCIs protect against these types of faults. If you are careful in these areas, there is less risk. Most home receptacles don't have GFCIs unless they are in damp locations (e.g. bathrooms, kitchen, outside). So I would focus on these areas first.

Here is a great article on why some appliances have 2- versus 3- prong plugs. Maybe it will help you assess your individual appliances and usage for risk.

https://www.quora.com/Does-the-third-ground-prong-on-electrical-plugs-improve-safety

GFCIs do not protect against circuit overloads. (Overloads may happen either from plugging too many things into the same circuit or from a short circuit.) That is what fuses and circuit breakers are for. They work regardless of the presence of a ground wire. As an owner of a couple of rental homes, I am as concerned about overloading the circuits through normal use (resulting in wires overheating in the walls) as I am about electrical faults (which may do the same thing, but it requires a failure versus a situation that may occur during normal day-to-day activities). If you have a lot of equipment that draws power (kitchen appliances, space heaters, amplifiers, etc.) you may want to figure out which receptacles are on separate circuits so you can spread out the electrical load. The fuses/breakers should protect you against overload, assuming the system was built/updated to code. But if you don't want to assume, you might want to think about how best to plug in all the stuff you have to distribute the load as evenly as possible.