Transformer for LED strip

You need a wallwart with an output that matches the voltage of the old one (Which may be AC or DC) and has a current rating at least as large as the one you're replacing. Polarity only matters for DC.

If your downlighters are halogen, the odds are that the transformer is just that, supplying AC, and if you use it to try and power anything requiring DC, it will probably destroy the latter.

Yes you need a bridge rectifier, and then the peak voltage may well be greater then expected, whether this is going to be true under load is difficult to say. Brian

Some of the colour changing LED strips work at 5V anything more will destroy them, so you really need to know what voltage your particauly strip needs. The original adapter should say oin it what V and I or power you need.