I am studying some aspects concerning string distance functions, and I am sure there are generic results available in the field of metric spaces, but I have not been able to find appropriate references.

The problem I have is the following. Let $d$ be a [string distance function][1]. The operation '+' denotes string concatenation. Consider the following trivial property:

$$d(a, b) = d(a+c, b+c) \tag{I} \label{eq1}$$

For example, under a function satisfying this property the distance between the strings ``abc`` and ``def`` would be the same as between ``abcxyz`` and ``defxyz``. Many of the most famous distance functions satisfy this property. For example, the famous [Levenshtein][2] distance does. However, not all of them do. For example, consider the following $n$-gram based distance (adapted from [this][3] paper):

$$d_1(a, b) = 1 - \frac{ \left| a_2 \cap ​b_2 \right|}{\mathrm{max} \left( \left| a_2 \right|, \left| b_2 \right| \right)}$$

where $a_2$ and $b_2$ are the sets of distinct $2$-grams in the strings $a$ and $b$, respectively.  For the strings``"abcde"`` and ``"abhij"`` $d_1($`"abcde"`$,$`"abhij"`$)$ is equal to $0.75$. In case we append the suffix ``"x"`` to both strings, the new distance would be $d_1($`"abcde"`$+$`"x"`$,$`"abhij"`$+$`"x"`$) = 0.8 > 0.75$. Conversely, in case we append the suffix ``"xy"`` to both strings, the new distance would be $d_1($`"abcde"`$+$`"xy"`$,$`"abhij"`$+$`"xy"`$) = 0.667 < 0.75$. Note that in the first case the value obtained is greater than the original (">"), and in this second case, the value obtained is less ($<$).

Of the distances that do not satisfy \eqref{eq1}, there are some of them that satisfy a more relaxed condition:

$$d(a, b) \leq d(a+c, b+c) \tag{II} \label{eq2}$$

For example, the following distance function, also based on $n$-grams, satisfies this property (adapted from [this][3] paper):

$$d_2(a, b) = \left| a_2 \ominus ​b_2 \right|$$

i.e. size of the symmetric difference between the two sets of $2$-grams. 

I have several questions regarding all this:
1. What is the name of property \eqref{eq1}?
1. What is the name of property \eqref{eq2}?
3. I am looking for the proof that if two distances $d_1$ and $d_2$ satisfy \eqref{eq2}, a new distance $d_3$ defined as a linear combination of them (i.e. $d_3(a, b) = \alpha \cdot d_1(a, b) + \beta \cdot d_2(a, b)$, for $\alpha, \beta \in \mathbb{R}$) will also satisfy \eqref{eq2}.

Thanks a lot in advance

  [1]: https://en.wikipedia.org/wiki/String_metric
  [2]: https://en.wikipedia.org/wiki/Levenshtein_distance
  [3]: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.409.9536&rep=rep1&type=pdf