Does Buddhism explain the past karma, and rebirth, of a short-lived fetus? When the rebirth-linking consciousness arises in the fetus (I read elsewhere at around 26 weeks), but the fetus dies (somehow) prior to or during birth, is it possible to understand why this kind of existence happened? And could it be a result of previous lifetime good or bad karma?  

In this situation, while there are senses, the fetus cannot make any conscious decision or thought processes or have any understandings yet. Of course depending on the situation physical suffering may have occurred. 

So I am trying to understand what kind of link this lifetime is, between the previous and next lifetime, and the effect on continuing rebirths.
is it possible to understand why this kind of existence happened?

not for a worldling
workings of kamma are inexplicable, and are not to be pondered upon lest they cause disquietude and agitation and because such pondering isn't conducive to attainment of nibbana, that's from the Buddha
workings of destiny are yet less explicable, if i may add, because destiny is a combination of personal kamma and fortuity

&quot;There are these four unconjecturables that are not to be conjectured about, that would bring madness &amp; vexation to anyone who conjectured about them. Which four?
&quot;The Buddha-range of the Buddhas is an unconjecturable that is not to
be conjectured about, that would bring madness &amp; vexation to anyone
who conjectured about it.
&quot;The jhana-range of a person in jhana...
&quot;The [precise working out of the] results of kamma...
&quot;Conjecture about [the origin, etc., of] the world is an
unconjecturable that is not to be conjectured about, that would bring
madness &amp; vexation to anyone who conjectured about it.
&quot;These are the four unconjecturables that are not to be conjectured
about, that would bring madness &amp; vexation to anyone who conjectured
about them.&quot;

Acinteyya sutta (AN 4.77)