For question #1, at least some US-based online brokers do permit direct purchases of stocks on foreign exchanges. Depending on your circumstances, this might be more cost effective than purchasing US-listed ADRs.
One such broker is Interactive Brokers, which allows US citizens to directly purchase shares on many different foreign exchanges using their online platform (including in France). For France, I believe their costs are currently 0.1% of the total trade value with a 4€ minimum.
I should warn you that the IB platform is not particularly user-friendly, since they market themselves to traders and the learning curve is steep (although accounts are available to individual investors). IB also won't automatically convert currencies for you, so you also need to use their foreign exchange trading interface to acquire the foreign currency used to purchase a foreign stock, which has plusses and minuses. On the plus side, their F/X spread is very competitive, but the interface is, shall we say, not very intuitive.
I can't answer question #2 with specific regards to US/France. At least in the case of IB, though, I believe any dividends from a EUR-denominated stock would continue to accumulate in your account in Euros until you decide to convert them to dollars (or you could reinvest in EUR if you so choose).