About understanding a Matched Filter I have some basic doubt in understanding a matched filter (MF). Please note that I have already seen very informative post in this context Understanding the Matched Filter

In what follows, I list three information about matched filter that I found in standard textbook and wikipedia:

(1) Wiki page says that MF is obtained by correlating a known signal, or template, with an unknown signal to detect the presence of the template in the unknown signal. 
In digital communication, I assume that the known template is the pulse shape p(t) used by the transmitter [e.g., 'sinc-like' impulse response of root-raised cosine (RRC) spectrum] and the unknown signal is the noisy received signal r(t). 
Assuming that the receiver knows p(t) used by the transmitter, the receiver convolves r(t) with p*(-t). The output of convolution is sampled at decision points. 

(2) A standard textbook on communication says that the receiver does a convolution of the received waveform with the time-reverse conjugate of channel response h(t). That means, MF operation consists of convolution of r(t) with h*(-t), followed by a sampler. 
(3) The textbook also says that MF can be seen as an equalizer. 

I understand that MF does a convolution followed by a sampling. That is it. Simple. But I am seeking detailed clarity in my understanding of the philosophy of this operation.

Questions:

(A). What is p(t) in the MF operation? Suppose I have NRZ data, then do I use the 'rectangular pulse of NRZ' or the 'sinc-like impulse response of RRC' as p(t) in the convolution? Wiki says, rectangular NRZ pulse is used. Why so? 

(B). Refering to the two different notions in (1) and (2), seems my understanding of the relation between p(t) and h(t) is unclear. Can someone make it clear with simple explanation?

(C). How do you explain MF operation as an equalization? Equalizer removes ISI, MF removes the effect of noise...

(D). What is partial MF?

I really appreciate your help in improving my understanding of this beautiful concept of matched filtering.
I just want to comment on the confusion between $p(t)$ and $h(t)$. What happens is that we transmit a signal $s(t)$ where

$$s(t)=\sum_ks_kg(t-kT_s)$$

where $s_k$ are the data symbols, $g(t)$ is the transmit pulse shape, and $T_s$ is the symbol time. 

Transmitting this over a wireless channel with a channel impulse response $h(t)$, the channel acts acts as a filter, and thus the channel output is the convolution between the transmitted signal and the channel, i.e.,

$$y(t) = s(t)\star h(t) = \sum_ks_k\underbrace{\left[g(t-kT_s)\star h(t)\right]}_{p(t-kT_s)}$$

So, the convolution of the transmit pulse shape and the channel is considered in the matched filter design.

By the way, the purpose of the matched filter is not to "remove the noise", but rather to maximize the signal-to-noise ratio (SNR) by maximizing the received signal power at the output of the matched filter.