How do mosquitoes maintain telomere length?
It seems that the current model is that recombination is the mechanism by which telomere length is maintained in the malaria vector Anopheles gambiae. Here is an exerpt of Roth et al paper that proposed the recombination mechanism:


  The insertion of a transgenic pUChsneo plasmid at the left end of chromosome 2 provided a unique marker for measuring the dynamics of the 2L telomere over a period of about 3 years. The terminal length was relatively uniform in the 1993 population with the chromosomes ending within the white gene sequence of the inserted transgene. Cloned terminal chromosome fragments did not end in short repeat sequences that could have been synthesized by telomerase. By late 1995, the chromosome ends had become heterogeneous: some had further shortened while other chromosomes had been elongated by regenerating part of the integrated pUChsneo plasmid. A model is presented for extension of the 2L chromosome by recombination between homologous 2L chromosome ends by using the partial plasmid duplication generated during its original integration.


You might also be interested in this paper: Genes required to maintain telomeres in the absence of telomerase in Saccharomyces cerevisiae and then look at whether there are mosquito homologs of the identified yeast genes that might be involved in telomere maintance.