Universal Turing Machines in "Computational Complexity" by Papadimitriou The first part of this question has been solved (see comments).

In the book "computational complexity" by Papadimitriou, a Universal Turing Machine is given. But this machine is not concrete, in the sense that it contains variable parameters. For example, the states of the input Turing Machine are represented as binary strings of fixed length. Hence, no matter how long we let this length be, there will always be Turing Machines with so many states that they cannot be represented. Can this machine then really be called a UTM, or at least a template for a UTM?

I also read that a 2-state, 3-symbol UTM has been discovered. This would mean that the action table contains only 6 entries. Am I right in my conjecture that such a machine must have a very complex encoding of the input TM?
In response to your second question, you are correct -- a 2-state, 3-symbol state TM was shown to be universal by then-undergraduate Alex Smith in 2007 by reduction to a universal cyclic tag system, which makes for a complex encoding.  Here's the original proof, which uses Wolfram's nonstandard notation for representing Turing Machines.

It's easy to intuit why simpler universal Turing Machines require more complicated encodings -- after all, the information involved in the computation needs to go somewhere.  For example, in the encoding scheme implicit in Smith's proof, we would first need to first encode the Turing Machine under simulation within a cyclic tag system, then use Smith's scheme to represent the cyclic tag system within the 2-3 Turing Machine.