Reading CGTF basis from Basis Set Exchange site I'm making a simple electronic structure code as an exercise, and I want to confirm that my understanding about basis set data from BSE is correct.
The CGTF of H in aug-cc-pVTZ basis is given as:
H    S
 33.8700000              0.0060680        
  5.0950000              0.0453080        
  1.1590000              0.2028220        

H    S
  0.3258000              1.0000000        

H    S
  0.1027000              1.0000000        

H    S
  0.0252600              1.0000000        

H    P
  1.4070000              1.0000000        

H    P
  0.3880000              1.0000000        

H    P
  0.1020000              1.0000000        

H    D
  1.0570000              1.0000000        

H    D
  0.2470000              1.0000000        

END
S-type GTF is simple, but I have questions about the number of other high-l GTFs.
P-type GTFs
I think the values
H    P
  1.4070000              1.0000000 

mean that $\chi = (1.00000) g_{p}$. Then Does that means there are three p-type function ($x$, $y$, $z$)?
D, F-type GTFs
If that so, in the upper case. How D, and F-type GTFs? Specifically, if we consider D-type GTFs, are there six gaussians ($x^{2}$, $y^{2}$, $z^{2}$, $xy$, $yz$, $xz$) or five gaussians following the atomic convention ($xy$, $xz$, $yz$, $x^{2}-y^{2}$, $3z^{2}-r^{2}$)?
Yes it is correct that for $p$-type basis functions that you have all three $p$-functions, i.e. $p_x$, $p_y$, $p_z$.

For basis functions of larger angular momentum, you are also correct that there is two "different" conventions. The cartisean coordinate and the spherical coordinate convention.

In many electronic-structure programs both conventions are used, since the integrals are calculated in cartisean coordinates and then afterwards transformed to spherical coordinates, i.e. as you stated for $d$-type functions going from 6 basis functions to 5 basis functions.

In regard to make you own electronic-structure code, I would recommend keeping everything in cartisean coordinates, because this will make your life simpler. (Keep in mind that if you compare you energy with from a Hartree-Fock calculation with other programs you will have a slightly lower energy, because you have 1 more $d$-type basis function).

For a simple electronic-structure program a very nice guide for molecular integrals can be found here, http://joshuagoings.com/2017/04/28/integrals/