orthogonal family and pulse shaping filter
In general for a pulse shaping filter and any orthogonal code family: No.

UPDATE: See @Engineer's answer which I confirmed by excluding the first row in the set of Walsh codes (the all 1 row) the remaining rows which have a balance of 1's and 0's are indeed all orthogonal.

*SECOND UPDATE: @MBaz has pointed out here that with Root-Raised Cosine filtering the entire code set will remain orthogonal: Orthogonal Codes for Band Limited Channel

A detailed mathematical proof along with the criteria where orthogonality can be maintained would be a much better answer, but since no one has submitted one I am at least providing here a simulation result that demonstrates a specific example of an orthogonal family where after pulse shaping with a raised cosine filter the orthogonality was not maintained. (The original signal can of course be extracted at the sampling locations with no distortion given the zero-ISI property of a raised-cosine filter, so that extracted signal would still be orthogonal after sampling, but the question was specifically on the orthogonality of the pulse shaped waveform itself ($g(n)*s_n(n)$)  

The formal definition for orthogonal signals is that their inner product (also called the dot product) is zero. 

Consider a simple case of the Hadarmard $H_4$ matrix, with each row forming a set of four orthogonal Walsh codes:

$\begin{bmatrix}1 &amp; 1 &amp; 1 &amp; 1\\1 &amp; -1 &amp; 1 &amp; -1\\1 &amp; 1 &amp; -1 &amp; -1\\1 &amp; -1 &amp; -1 &amp; 1\end{bmatrix}$

Before pulse shaping it is easy to confirm orthogonality, for example between row 1 and row 2 using the dot product of index by index product and sum:

$$(1\times1)+(1\times-1)+(1\times1)+(1\times-1) = 0$$

And likewise betweeen row 1 and row 4:
$$(1\times1)+(1\times-1)+(1\times-1)+(1\times1) = 0$$

The result of a simulation of this set with a raised cosine filter suggests that orthogonality is maintained in the case of the first two rows, but is not between row 1 and row 4. The resulting waveforms from the first two rows after upsampling by 10 and pulse shaping with a raised cosine filter are shown in the figure below (the sample locations for decoding the original signal are at samples 50, 60, 70 and 80): 



The computed dot product for these two waveforms was $4.4E-16$. (Close enough to zero so orthogonal).

Row 1 vs Row 3 is similarly quite low at $9.1E-16$ (again orthogonal). 

However
Row 1 vs Row 4 was $-0.465$ which is a very strong correlation and clearly no longer orthogonal. 

The plot showing the pulse-shaped Row 1 and Row 4 is shown below. 



The dot product is the sample by sample product and then the sum (or integration) of the result. Plot of the product of each of the waveform pairs gives further insight into the loss of orthogonality. Note how the product of the pulse-shaped row 1 and row 2 is very symmetric with similar positive and negative areas under the curve, while the product or the pulse shaped row 1 and row 4 has a significantly larger negative area, thus destroying orthogonality in that case: