Yes, and yes.
Climate affects diatoms in complex ways. As the planet warms due to the increase in carbon dioxide, scientists predict that larger marine plankton, like diatoms, will decrease compared to smaller plankton, like coccolithophores and cyanobacteria. In lakes and rivers, a changing climate alters river flow in many parts of the world. The frequency and severity of droughts and floods is changing, which impacts diatom species and where they grow. Furthermore, climate controls circulation patterns and thermal stratification of lakes and oceans, which alter diatom species composition.
Diatoms affect climate on a global scale. As diatoms photosynthesize, they "breathe" in carbon dioxide from the atmosphere and they "breathe" out oxygen. Although diatoms are very small, they live in the vast oceans, the world over. The effect of fixation of carbon by diatoms and release of oxygen alters the chemistry of the atmosphere.
One recent model of ocean biogeochemistry explored the effects of climate change on diatom distribution (Bopp et al. 2005). The results suggest that climate change leads to more nutrient-depleted conditions in surface waters of the ocean. As a result, small celled phytoplankton are favored and diatoms decrease in abundance.