How to separate and analyse a sample of cations of alkaline and earth alkaline cations and ammonium?
In a qualitative analysis, the fast way is always the wrong way. There are already so many issues with the "long" version that I am surprised that this is remotely successful.

The cations of groups 1-3 are always the most difficult to separate and during my time as a teaching assistant were by far the biggest source of error.

The proper course of analysis is 


dissolve sample (completely)
precipitate ion
dissolve precipitate
specific reaction
repeat


The reason for separating the cations in the first place is, that they often act in similar ways and the reactions are not very specific. Certain cations should be tested multiple times, too.

All the reactions you are performing are equilibria and very dependent on the overall concentration. None of them form a precipitate quantitatively. Sometimes you might have problems recognising precipitates in the first place.

You need to perform double blind tests for all used chemicals, since already small contaminations may lead to false positives.

The main issue with your proposed strategy is that you will not precipitate aluminium with this strategy, since this is extremely pH dependent.
$$\ce{Al^3+ + 4OH- &lt;=&gt; Al(OH)3 + OH- &lt;=&gt; [Al(OH)4]- }$$
Without a buffered solution at around pH 9 and ageing the solution, there should not form a precipitate. I learned this by adding ammonium sulfide first and raising the pH slowly with ammonia. (This will co-precipitate cobalt, nickel, manganese, zinc, and iron [as sulfides], and chromium [as hydroxide]. This might be useful if you have to analyse for more ions later on.)
If a precipitate forms you can dissolve it with sodium hydroxide and use cobalt(II) nitrate to form Cobalt blue.
You can also test the presence of aluminium with morin as it forms a fluorescent complex with it. Note that $\ce{Mg(OH)2}$ is slightly fluorescent, too.

I cannot remember precipitating magnesium with that strategy in any sufficient amount. I would prefer precipitating it as an ammonium phosphate, which you can try to (re-)crystallise as struvite and look at them with a microscope.
$$\ce{Mg^2+ + NH4+ + PO4^3- &lt;=&gt; MgNH4PO4 v -&gt;C[H2O] MgNH4PO4.6H2O}$$

If you want to precipitate out aluminium and magnesium (and almost every other heavier element) you can use oxine, $\ce{C9H7NO}$ (abbr. $\ce{Hox}$), solution.
$$\ce{ M^{$x+$} + $x$\,Hox &lt;=&gt; [M(ox)_{$x$}] + $x$\,H+ }$$

Then precipitating calcium is straight forward with carbonate solution. Other earth alkaline metals will precipitate, too. You can dissolve this precipitate with acids or you can use the solid to perform flame tests. Since there can only be calcium, strontium, or barium, this should be conclusive. (Of course you could precipitate barium sulfate.)

The main pitfall with your proposed strategy is, that you will not quantitatively precipitate calcium hydroxide and it is as mentioned above quite concentration dependent. The solubility constant is not too small, i.e. $K_\mathrm{sp}=5.5\times10^{-6}$.

If you have precipitated everything so far, then the last thing left over are ammonium, potassium, sodium(, and lithium). The biggest problem is, you cannot mask them properly and you cannot precipitate them. You can use flame tests to determine their presence, but as long as you have sodium in it you might not see anything else. You should use a prism, or a spectrometer to determine the characteristic spectral lines. Apparently phones can do it now, too. Also see publiclab.org... makes you think why separate at all.

You will need to analyse ammonium from the sample. You can add sodium hydroxide, which releases ammonia. You can detect the presence of ammonia in the sample by holding a wetted universal indicator paper over the sample. Additionally, you should be able to smell it.

In other words ...
... be patient with your analysis as there are more pitfalls, false negatives, false positives, contaminations. Don't use short cuts, except atomic emission spectroscopy.