All these words can be used more generally and applied to atypical situations, so in principle each word could be used for everything else. But normally the core meanings apply, at least to some extent. Where they don't, it may be for historical or strange technical reasons, or because the organisation wants to mislead the public. Therefore I will only describe the typical uses.
An association is an organisation in which natural persons or sometimes other organisations can be members. Its members are typically connected by a common purpose or interest. E.g. the members of a professional association share the same profession. Sometimes membership in an association is obligatory for a certain class of people. Typically, most members have never seen each other. Occasionally there may be general meetings in which ideally all members should take part, but normally most don't bother. These meetings are not important for how the public or the association's members see the association. This is an important difference from the other two notions.
An assembly is a large body of people who meet in a suitable hall to get some specific things done in a few hours, a few days, or even a few years. (If it's longer than a day, they obviously go home or to their hotels in between. If it's as long as a year or longer, they also have vacations.) The purpose can be, e.g., electing a president, drafting a constitution for anything from a club (constitutive meeting, founders' meeting) to an entire country (constituent assembly, constitutional convention), or making a country's legislation (national assembly, parliament). To actually get things done, most assemblies have some ground rules for their operation, offices such as a speaker or assembly president, and subdivisions such as factions and committees.
A committee is a small, ideally expert, group of people whose task it is to prepare certain technical decisions by certain more important deciders (e.g. industry committee, government committee) or by an assembly (e.g. parliamentary committee). The members of the committee have been chosen or delegated by whoever it is that ultimately decides. The members all know each other well, since they are working closely together on highly technical problems. They may or may not meet often as a group, but if they do, a normal sized meeting room is sufficient.
As to the more concrete questions: Yours is a clear case of an association and certainly not a committee or assembly. The normal usage for association names is Association of [description of members] or Association for [field of interest]. Therefore Association for Higher Education and Disabilities in the USA is the best among your three choices.