a) Well, the first process is a MA(1) process, and MA processes are always stationary, no matter what.
The second process is an AR(1) process, and they are either stationary or not. This particular AR process is called a random walk process and is not stationary.
b) An I(1) process is a non-stationary process that has ben first differenced (excuse my English) in order to make that particular process stationary. You could test this with a Dickey-Fuller unit root test.
c) If two series are cointegrated then there exists some long-term equilibrium between those two series.
d) It can for example come about because of arbitrage opportunities that arise in markets.