Variable power supplies are rare because most loads are expecting a fixed voltage. Does the existing power supply have a control on it?
LED replacements may not dim in the way that you are expecting. You definitely need all the lamps to be the same, and I have bought LED lamps that differ when supplied in one batch. I seen (well, I've even designed and built) LED lamps that have constant brightness over wide voltage ranges, so if you get ones like that, dimming isn't possible.
For the connectors, I suggest you look at the Wago range. Just note that some of those, generally the ones without a release lever, aren't suitable for stranded wires, only solid.