Can you really make a new dinosaur from old DNA?

Unfortunately no dinosaur DNA has ever been extracted. There was some suggestion that some had been found in the past but it turned out to be due to contamination. Any DNA that might be found would probably be extremely degraded (broken up). To be able to produce a healthy organism you have to have all the genes in its genome. Genomes for advanced creatures tend to be in the order of billions of base-pairs, and the chance of extracting more than a few tens or hundreds of bases is just about zero. Even if we manage to find lots of DNA there is a large chance that most of this DNA will be junk (in higher animals about 90%+ of the genome is non-coding DNA). So unfortunately there is not really any chance of being able to bring dinosaurs back to life.

For an insect to be able to pass on the dino-DNA when they mate is also extremely unlikely. If we are talking about mosquitoes only the female drinks blood the males normally drink fruit and plant juices. When insects mate their mouth parts rarely come into contact and the dino-blood would never get near the reproductive organs.