Latitude Geographics developed Geocortex Optimizer for just this purpose. It's a program that installs as a service, and performs periodic monitoring of your ArcGIS Services and the servers behind them (by performing ping requests, web requests, map requests, and also by monitoring log files and performance counters).
There are also API hooks that let you connect a web viewer to the Optimizer collectors, so you can get info on how your web viewer is being used. (what extents, what tools, user activity, etc)
The data it collects gets pushed into a database and there is a reporting module that analyzes the data and presents the results as a web page. There are graphs and heat maps to help visually represent trends and usage. You can also have some reports emailed periodically.
DISCLAIMER: I work at Latitude Geographics, although not directly with the Optimizer product.
SDE is a middle tier software that translates calls from ESRI, or other, clients into SQL that is appropriate for a certain database. If you are connecting to a geodatabase, then SDE also manages a certain amount of metadata that is stored in that database's SDE or dbo schema. What is ArcSDE
For Vector data on SQL Server 2008, there are actually spatial types provided by Microsoft which you can be used to store your data. But by default, SDE will create tables using a storage type called SDEBinary or just Binary. This is basically storing the geometry as binary data in a side table (F-Table), and maintaining a pseudo-index table to improve performance (S-Table). I call it a pseudo-index table because it isn't an index at the database level.
I don't think Microsoft has provided a type that can store rasters in the database (other then a BLOB). SDE can add an ST_Raster type, but that type will not be added by default.
When you run the post-install program, or the enable or create geodatabase GP tools at later releases, you are adding the tables in the SDE or dbo schema that track feature classes and tables, along with other functionality such as versioning and archiving. Additional tables are added by the Geodatabase to track feature database, subtypes, topologies, etc.
The difference between direct connect(2-tier) and application service connection(3-tier) is where the SDE process is running. That is, where the translation between the SDE calls from the client and the DBMS calls from SDE are actually being translated. With a direct connection, that process is running within the client application (ArcMap, ArcCatalog, ArcIMS, etc). With an application service connection, that same process is started by a separate service called the giomanager. This giomanager can run on any machine, but is commonly run on the same machine that is hosting the database. An Overview of ArcSDE Geodatabase Connections
So, to answer your questions:
Question1: It depends on how you are connecting. If in your connection dialog box you are entering a port number into the instance field, you are running application service connection. If you are entering something like 'sde:sqlserver:myHostName', then you are running a direct connection. Creating spatial database connections - Scroll down to 'Adding a direct connection to a geodatabase in SQL Server'.
Question2: It depends. There are many things that can cause performance issues for both SQLServer, DBMS and file based rasters. But in general, file based rasters outperform those stored in databases. This is because the architecture of a database, and ArcSDE, isn't really designed to store and retrieve giant chunks of binary data. Similarly, large and externally complex vector data does not perform well either. Rasters stored locally will outperform those stored remotely. But all else being equal, I would expect that a raster stored in a remote file geodatabase to perform more slowly then one stored in an SDE instance, which would perform slower then one stored as a tiff in a remote directory, which would be slower than one stored in Image Server. Geodatabase and ArcSDE Raster Basics
Best Answer
Try to find out one or more of these criteria to determine if ArcGIS Server ist installed
There is a Process ArcSOM.exe and one or more ArcSOC.exe running on your machine
Under %PROGRAMFILES(x86)\ArcGIS* you'll find an installation of ArcGIS Server