By Caroline Valetkevitch NEW YORK (Reuters) – The euro hit its lowest level against the U.S. dollar since June 2020 on Tuesday and the Russian rouble was down in volatile trading as Russia’s invasion into Ukraine intensified and oil prices surged. The U.S. dollar index, which measures the greenback against a basket of currencies, jumped and was last up 0.6%, as investors flocked to safe-haven bets. Investors were on edge over the latest Ukraine developments. Russia warned Kyiv residents to flee their homes, and Russian commanders shifted tactics to intensify the bombardment of Ukrainian cities…