What exactly is the ‘online’ in OLAP and OLTP

To answer your questions in order:

1. The cube doesn't store medians, modes (or even averages), but you can write queries that calculate them and embed them as calculated measures in the cube. The ability to embed this sort of computation is one of the main unique selling points of OLAP technology.
2. If you have a dimension that can identify individual rows (which could be a degenerate or 'fact' dimension derived from an identifier on the fact table) then you can do a query based on individual rows. However, OLAP works in terms of dimensions and aggregates, so you would need to have a dimension capable of identifying individual rows (with an aggregate composed of one value).
3. Any OLAP tool can do what's described in (2), plus they generally support a mechanism known as 'drill-through' where the cube will return a bordereaux of the transactional data underlying a given slice that you drill through into.

If you want to do calculations that aren't directly available in the cube script, many OLAP tools such as the late, lamented ProClarity will allow you to formulate queries involving custom MDX based calculations. Unless the cube doesn't have the information you need to do the actual calculations, the custom MDX calculations should be able to support any computation you need.

Although OLAP queries are traditionally associated with statistical queries in aggregate, if you have a dimension that allows drill down to detail you need it is certainly possible to formulate queries that will calculate medians, percentiles or histogram queries from which modes can be inferred or computed.

For example, this has an example of a pareto analysis query, which is based on rankings.

Many cube products can operate in a hybrid or relational OLAP mode where they do not persist the data themselves but query it from an underlying database. In addition, pure ROLAP tools such as Business Objects, Report Builder or Discoverer can query from an underlying database and do work row by row. However, they tend to lack the sophistication of dedicated OLAP products, and they don't have much in the way of statistical analysis capability out of the box.

With some offline conversations, Zane was able to determine that if they could keep a connection open, it would hasten the validation.

To that end, I suggested they change the property on the Connection Manager to flip the RetainSameConnection property to True from its default.