How do aggregations work in SSAS

Sounds like you're getting orphaned fact rows a.k.a. early-arriving facts, meaning you have facts records that don't yet have a corresponding dimension member in one of the dimensions.

There's not really a way to solve this with the approach you're using because it's sounds like you'll always been a in situation where your dimensions are lagging your facts. But you can work around it by changing your dimension key error options in your fact processing. There is an option to ignore dimension keys not found. What you want to do is set that to "Ignore" and you might have to play around with some of the error limit options as well.

This should do the trick. Not the prettiest of solutions, but it will work and prevent these outages.

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.