As the voltage in the states is lower than that in the UK there will be no problems with insulation.
Theoretically true but unlikely to be a problem in practice for an ordinary domestic lamp. The smallest standard UK flex size would be fine for up to 330W on a 110V system.There may be a problem with cable size as the current for the same wattage lamp will be double that of the same wattage in the UK.
Lampholders are likely to be an issue indeed. AIUI not only do they not use bayonet in the US, they also use a slightly difference size of edison screw.If the lamp holders are bayonet you may have to change them to screw in as ( from what I recall ) 110 volt bayonet lamps are not easy to buy in the USA.
Good information.If you are altering plugs on other items that you take to the states remember that in USA power leads white is NEUTRAL and black is LIVE