groups, levels and denominator dof in mixed effect models

I am trying very hard (I am not a statistician) to understand the concepts of “groups” and “levels” in mixed effect models. In particular, I am trying to understand this in the context of the denominator degrees of freedom of the fixed effect terms of a model. I know that denDF is an open question in mixed effect models, so for the sake of the current discussion I would refer to the estimated denDFs based on the inner-outer rule proposed by Pinheiro and Bates (pag. 91) and summarized here.

First off, can anyone define what “levels” and “groups” are in mixed effect model? My understanding is that groups are defined by the structure of the random effects (and therefore by the model). However, I am confused with the definition of levels. Also, is there any relationship between levels and groups? In other words, can I modify the number of groups (by changing the structure of the random effects) independently on the levels. Are levels inherently defined by the structure of the data, or they emerge from the definition of the model?

Regarding the definition of denDF. Pinheiro and Bates define:

$denDF_i = m_i -(m_{i-1}+p_i), i=1,…,Q+1$

where $m_i$ denotes the total number of groups in level $i$, $p_i$ denotes the number of degrees of freedom corresponding to the terms estimated at level $i$, and $Q$ is the number of grouping factors. My questions are:

1. This definition is highly dependent on the definition of levels. How do I know what groups belong to what level?

2. According to this definition, there seems to be $Q+1$ denDFs. But I can change the number of grouping factors by changing the structure of the random effects, while I always need as many denDFs as the number of fixed effect terms to be estimated plus 1 (the intercept). For example, in the Machines example from the book, I could define random=~1|Worker (Q=1, hence 2 denDFs) or random=~1|WorkerMachines (Q=2, hence 3 denDFs). Wouldn’t I need only two denDFs in both models, Intercept and Machine? Why does this depend on the definition of the grouping?

3. I cannot understand the logic of this formula. Especially, why do I need to subtract $p_i$?

Finally, is there a rule of a thumb to figure out how to define groups (and therefore random effect structure) in a model? This seems to be a very important design decision: the definition of the groups influences the denDF, and therefore the results of the hypothesis tests on the fixed effect terms.

I do apologize for my naive questions, but I am really trying to understand this. If this is too basic for this forum, can you give me specific references where I can read about these issues? Thanks a lot for your help.