LMS and delta impulse response- Equalization concepts
You're right. LMS equalizer uses a known input to minimize the error. For communication purposes, this is either provided by a training sequence, or in a decision directed mode, the detector decisions are fed back as known data.

The delta function is also correct. Suppose that the channel impulse response is $h(t)$ and frequency response $H(f)$. Then the equalizer should ideally be the inverse of this function in frequency domain (remember the convolution in time domain and hence multiplication in frequency domain).
$$H(f)\cdot \frac{1}{H(f)} = 1$$
and its inverse Fourier transform is
$$F^{-1}(1) = \delta (t)$$

The fourth tap being 1 implies that there is a delay of 4 symbols with respect to your time base at the Rx. Finally, to further verify your results, plot the frequency responses of the channel and the equalizer response, and you will see an approximately inverse relationship.