If I don't need a primary key should I not add one to the database?
If you don't need a primary key then don't use one. I usually have the need for primary keys, so I usually use them. If you have related tables you probably want primary and foreign keys.
A primary key will always help with query performance. So if you ever need to query using the "key" to a "foreign key", or used as lookup then yes, craete a foreign key.
Yes, but only in the same sense that it's okay not to use a seatbelt if you're not planning to be in an accident. That is, it's a small price to pay for a big benefit when you need it, and even if you think you don't need it odds are you will in the future. The difference is you're a lot more likely to need a primary key than to get in a car accident.
You should also know that some database systems create a primary key for you if you don't, so you're not saving that much in terms of what's going on in the engine.
A primary key uniquely identifies a row in your table.
The fact it's indexed and/or clustered is a physical implementation issue and unrelated to the logical design.
You need one for the table to make sense.
I don't know. I have used a couple tables where there is just a single row and a single column. Will always only be a single row and a single column. There is no foreign key relationships.
Why would I put a primary key on that?
A primary key is mainly formally defined to aid referencial Integrity, however if the table is very small, or is unlikely to contain unique data then it's an un-necessary overhead. Defining indexes on the table can normally be used to imply a primary key without formally declaring one. However you should consider that defining the Primary key can be useful for Developers and Schema generation or SQL Dev tools, as having the meta data helps understanding, and some tools rely on this to correctly define the Primary/foreign key relationships in the model.
Well...
Each table in a relational DB needs a primary key. As already noted, a primary key is data that identies a record uniquely...
You might get away with not having an "ID" field, if you have a N-M table that joins 2 different tables, but you can uniquely identifiy the record by the values from both columns you join. (Composite primary key)
Having a table without an primary key is against the first normal form, and has nothing to do in a relational DB
No, unless you can find an example of, "This database would work so much better if table_x didn't have a primary key."
You can make an arguement to never use a primary key, if performance, data integrity, and normalization are not required. Security and backup/restore capabilities may not be needed, but eventually, you put on your big-boy pants and join the real world of database implementation.