Do the primers used for Covid-19 test work with all new strains?
I TRY to answer. First of all, no we cannot be 100% sure that the primers are effective against all strains.
However, the strains that are under media attention right now are due to variants in the spike protein, while RT-PCR primers are designed on different genes.
One WHO document (I do not know if it's the official guideline) targets RdRp (RNA-dependent RNA polymerase) and a gene called E gene.
While the document itself nor the paper on which it is based explicitly addressed conservation of the target genes, I found evidence for relatively high conservation of at least RDRP in this paper.
Personal opinion here: in looking for RT-PCR primers, you always need a balance between targeting genes that do not vary in the species, and targeting genes that may aspecifically amplify in different species (e.g. SARS-CoV).
Final answers:

how can we be sure that there are not new strains with sequences not complementary to the primers?

We cannot. However, as far as the variants media are talking about, we know that RT-PCR has some ability to detect them, since they have been sequenced as a consequence of some patient testing positive to RT-PCR.
We cannot exclude the existence of unknown strain. They could be identified by e.g. sequencing patients with COVID-2 symptoms testing negative to RT-PCR (and possibly testing positive to the antigenic test). I don't know if such cases have been observed.

Does the current covid19 test target highly conserved RNA regions?

Yes, the current RT-PCR based tests target conserved genes (although I wouldn't call the highly conserved, because of the caveat I expressed above as a personal opinion)

If that is correct, how do we know that those regions are highly conserved?

Mostly by sequence similarity. Regions that have the same sequence in all the individuals of a species and in related (or distant) species are conserved.