ASK and Manchester Encoding
I would suggest you first demodulate your signal. Take the received signal y[t] = x[t]*cos(2*pi*1000*t) and multiply it by cos(2*pi*1000*t). Remember, $ cos(A)cos(B) = 1/2 (cos(A-B) + cos(A+B)) $. You'll be left with x[t]*0.5*(cos(0) + cos(2*pi*2000*t) = x[t]*0.5*1 + x[t]*0.5*cos(2*pi*2000*t). Filtering the remaining signal with a low pass filter at 1k Hz, leaving only x[t]. From there you can try and answer the rest of your questions.

In Matlab this will look like:

t = 0:1/Fs:END_TIME;  % Fs is your sampling rate. This makes a vector of individual time ticks
y_demod = y.*cos(2*pi*1000*t);   % This multiplies every element in Y with the cos(...) 
                                 % ... element. Make sure they're the same size.
a = (1/Fs)*(Fs/4);               % A cutoff frequency of approximately 1/2 the sampling rate
x_est = filter([1 1-a],a,y_demod);


The simple 1-pole low pass filter was taken from here: https://stackoverflow.com/questions/1783633/matlab-apply-a-low-pass-or-high-pass-filter-to-an-array