The most idiomatic choices of preposition are:
- 65 years and over
- 65 years and above
- 65 years plus (maybe also styled as '65+')
- 65 years and older
'Greater' and 'higher' are simply just not idiomatic when speaking about age, although you can use 'higher' when referring back to the group (eg "the higher age groups"), it just isn't natural to call the group by that.
Which you use from among the idiomatic choices is up to you, unless you are working with a style guide.
As to the question of 'or/and' - this depends on whether you are presenting an option or not. For example:
- You can enter the bar if you are 18 or over.
- This bar is for those aged 18 and over.
In example 1, an individual ('you') is referred to. They can only be one age, so they must be 18 or over. Example 2 refers, not to an individual, but to anyone that can enter the bar - those aged 18 and those aged over 18. Your specific example is referring to a group, so use 'and'.