Yes. So, the higher the speed
Yes. So, the higher the speed, the higher the energy consumption and thus range goes down. This is true for any vehicle - EV or gas. EPA ratings are based on a given speed limit - 65 mph. If you travel at 80 mph your effective range is worse. Conversely, if you're travelling around a city, the EV usually has a higher range and some of the impact of stop-and-go traffic is offset by the regen function. EVs range tends to be higher in the city as a result.
Charging on the other hand is not impacted by speed. That said, cold temperatures can impact the rate of charge if the pack is cold, and thus EVs heat the pack in order to optimize charging. Current EVs are susceptible to the cold as some of the energy stored in the pack must go to warming it up, so again range is impacted. Heat pumps go some way to reducing the impact of the cold on range. If/when solid-state batteries come about, temperature has less impact on these batteries. However, at this point, they're comparatively 1) more expensive, 2) insufficient storage compared to what's currently being used - Lithium-ion.