Is it always required to achieve stationarity before performing any time-series analysis? For example, I know that for ARIMA models stationarity needs to be achieved. What about Exponential Smoothing? Is it also required?
 A: ARIMA models are not stationary, ARMAs are. ARIMA includes the integration terms, e.g. a random walk model is ARIMA(0,1,0) and it's not stationary.
There's a couple of different ways to exponentially smooth, here's EWMA and a different version. Neither of them requires stationarity.
Here's an example in MATLAB with fitting ARIMA(0,1,1) into S&P 500 index in 2014, and smoothing it. You can see that the series are clearly non-stationary, and both approaches work fine.
d=fetch(fred,'sp500','1-jan-2014','31-dec-2014');
y = d.Data(:,2);
y=y(~isnan(y));
dt = d.Data(~isnan(y),1);

subplot(2,1,1)
plot(dt,y,'r')
datetick

% arima(0,1,1)
fitA=estimate(arima(0,1,1),log(y));
R=infer(fitA,log(y));
hold on
plot(dt,exp(log(y)-R),'k')
legend('Actual', ...
       'ARIMA(0,1,1)', 'location','best');
ylabel('S&P 500');
xlabel('Date');
title('S&P 500 Smoothing');plot(dt,y,'r')

% exponential smooth
alpha = 0.45;
ewma = ones(size(y));
ewma(1) = log(y(1));
for i=2:length(y)
    ewma(i) = alpha*log(y(i))+(1-alpha)*ewma(i-1);
end

subplot(2,1,2)
plot(dt,y,'r')
datetick
hold on
plot(dt, exp(ewma),'b');

legend('Actual', ...
       'Exponential Smoothing', 'location','best');
ylabel('S&P 500');
xlabel('Date');


A: exponential smoothing models do not assume stationary data.  
Citation: see Hyndman and Athana­sopou­los:
"every ETS [exponential smoothing] model is non-stationary"
A: Exponential Smoothing model is a particular form of an ARIMA model.  Instead of identifying the patterns, outliers and trends, you assume them.  
A: Structural time series models do not assume stationarity either.  
These models a la Andrew Harvey are estimated via Kalman filter type of algorithms. 
http://www.stat.yale.edu/~lc436/papers/Harvey_Peters1990.pdf
A: As it stands, I do not even agree that stationarity "needs to be achieved" before performing time-series analysis with an ARIMA process. It needs to be clarified what the goal of the analysis is. 
For example, there are tons of posts on this site relating to testing the null $\rho=1$ in the model $Y_t=\rho Y_{t-1}+\epsilon_t$, which, under the null, is an ARIMA(0,1,0) model, i.e., the Dickey-Fuller test. This shows that it is possible to do hypothesis testing in ARIMA models.
As another example, take the DF distribution itself:
$$
T(\hat{\rho}-1)\Rightarrow\frac{W(1)^2-1}{2\int_0^1W(r)^2d r},
$$
where $W$ is standard Brownian motion. To be sure, a "non-standard" random variable, but one which, for example, shows that $\hat{\rho}-1=\mathcal{O}_P(T^{-1})$. Thus, OLS consistently estimates the parameter of the process - even at a faster than the usual $\sqrt{T}$-rate!
On the other hand, if you mean that things like the central limit theorem or a law of large numbers are required to work, then, yes, stationarity may be important.
