In the periodic table, why doesn't the 2nd row have exactly 2 elements?
The pattern is better expressed this way:

Row 1: 2 elements
Row 2: 2+6 elements
Row 3: 2+6 elements
Row 4: 2+6+10 elements
Row 5: 2+6+10 elements
Row 6: 2+6+10+14 elements
Row 7: 2+6+10+14 elements


The reason comes down to how the electrons fill the available energy levels.  The thing that differentiates one element from another is the proton number, and each time a proton is added and a new element defined it requires one more electron to neutralise the charge.  That electron naturally occupies the lowest available energy level in the atom.

The energy levels available are defined by the quantum state, including the quantum numbers $n, l,$ and $m_l$.  $n$ is the principle quantum number and relates to the period the element is in, or the shell.  $l$ is the angular momentum quantum number which defines the sub shell s, p, d, f, of which there are $n$ subshells whose values are $l=0,\dots {n-1}$.  The magnetic quantum number $m_l$ further subdivides the subshell into orbitals, of which there are $2l+1$ orbitals whose values are $m_l=-l,\dots {+l}$.

subshell                 number of orbitals    subshell
label     l value       (number of ml values)  electron capacity
 s         0               1                   2 
 p         1               3                   6
 d         2               5                   10
 f         3               7                   14


Each orbital (i.e. each value of $m_l$) may contain 2 electrons  The available quantum numbers are:

n    l values   (subshells)   ml values        total shell electron capacity
1    0           (s)           0                 2          
2    0,1         (s,p)        -1,0,1             2+6        
3    0,1,2       (s,p,d)      -2,-1,0,1,2        2+6+10     
4    0,1,2,3     (s,p,d,f)    -3,-2,-1,0,1,2,3   2+6+10+14  


The pattern is there, but it doesn't seem to match up because by row 7 you would have 98 electrons in the shell and might expect the row to contain 98 elements.  This is not the case!  The energy levels for the $l$ values with large deviations from $0$ (i.e. orbitals with high angular momentum) become increasingly far apart, so even though 3d orbitals exist in the third row, they are so much higher in energy than the 3p orbitals that they are higher even than the 4s orbitals.  This happens again in row 4 where 4f orbitals are so much higher in energy than 4d orbitals, they are even higher than 5s.

The different in energy is not always so large in specific cases, so there are examples where bonding or ligands or symmetry cause the energy levels to switch around, but otherwise the trend is true. If they can produce (or discover) some elements in the next row, we might expect to see the 5g shell finally start to get filled and have a new block in the table, but realistically it doesn't look like they will be stable atoms.

I think I'm missing the reason why we arrange the elements into this table in the first place.  The periodic table is arranged so that all elements in the same column have the same number of outer electrons.  This is useful because elements with the same outer shell configuration of electrons react in similar ways, so they are grouped together in columns. This has the effect of creating 'periods' that begin with a reactive metal and end with an inert gas. Row 1 has two elements, and it's special in a way because it's the first in the series.  Hydrogen has 1 electron, and a maximum capacity of 2 electrons.  This means it behaves in a similar way as other elements that only have 1 electron in their outer shell (group 1), and also in a similar way to elements that only need 1 more electron to complete their outer shell (group 7).  Where do we position it then?  Often it is placed somewhere in the middle.  Helium then has a full outer shell, which means it has many similarities to other elements with a full outer shell and goes in group 8 (or group 0) on the right.

Edit:
The alternative period table that you mention, the left-hand version, is laid out so as you read it from left to right you fill up the orbitals.  This takes in to account that situation I describe where 3d orbitals are higher than 4s, and using this left-hand table you clearly see that. However this alternative table does not allow you to easily read the valance shell configuration like the Mendelev one does.  Other types of table highlight different kinds of properties or relationships, which can be more useful in certain situations.  The labels on the left (1s; 2s; 2p 3s; etc) relate to the outer subshell in that row, so Carbon is in the "2p" section of that chart while Magnesium is in the "3s" section on the same row.