Your second question is the answer to your first question: The dots indicate the head node of the respecitive phrase.
butis the head of the entire sentence,
lookis the head of the VP
is is the head of the relative clause
from which their name is derived.
In case a sub-tree consists of a single node only, heads are not marked by a dot as there is only one node that could be the head anyway - as in the subject NP
chandeliers is the only node and at the same time the head node of the NP
chandeliers. If it had any dependents, e.g. a determiner, it would look like just like the structure around
So you can identify that
but must be the head of the sentence that
chandeliersbelongs to because
butis the node that is marked by a dot. It can not be
do because the are not marked by a dot (more precisely, they are, but on a different level:
look is the head of the VP
look great, where
look is the head and
great is its dependent, and
do is the head of the VP
nowadays do not ...; however, as indicated by the lines being more rightwards, this is nested one level deeper than the main sentence, and therefore,
do are the head of the respective VPs rather than of the entire sentence, because otherwise the dot should appear on the leftmost level).
If you want, you can just rotate the tree 90° and add arrow tips:
This graph tells you the exact same thing as the output in the Watson format. It just looks a bit differently - this is really no more a matter of style.
Your question is thus not actually one about English slot grammar in particular: It is about the graphical representation of dependency trees - where the Watson parser referred to in the paper happened to use some less common format for displaying the output, which is, however, independent of the type of dependency grammar (e.g., slot grammar) or language (e.g, English). A universal Stanford dependency parse could be displayed in the way your Watson SG screenshot does, and a slot grammar analysis tree could be displayed in the way you are used to from the literature (like in the graph I provided above); the difference lies only in whether to show the graph horizontally or vertically, or whether to choose dots or arrow tips for the edges and apparently a vertical tree with dots happens to be more useful for the three column display so they chose that format to display the same thing.
As a side note, you should be a bit careful about wording: head ≠ mother. The paragraph talks about heads and modifiers, which is a notion that involves some semantic/linguistic information, while "mother" is a purely syntactic (graph-theoretic) concept that is about which node is immediately dominant to some other node.
but is not the mother of
chandeliers - it is the head of the phrase that is the mother of the node consisting of
chandeliers. The mother node would be the complex node consisting of the entire sentnce; or in the NP
their name, the mother node of both
name would be the complex node consisting of exactly these two nodes.
But since dependency trees, as opposed to constituent trees, are not direclty interested in constituent hierarchies and relations such as immediate dominance or sisterhood, that the complex consituent is the mother node is not directly of interest here - what is more important is that
but is the head of the phrase that
chandeliers belongs to.