Okay here's a few for the experts.

I've been a Tamiya purist ever since the supply of Gunze Aqueous seemingly dried up here in Canada.

But I've just bought some Vallejo Model Color (only proper color matches I could find for a subject) and some Model Air.

Obviously I don't need to thin the Model Air, what's the best thinner for Model Color? Can I use alcohol? I hate buying proprietary thinners.

Second, what's the best way to apply Vallejo pigment?

From my understanding the Vallejo Thinner is the best for Model Color, but I have heard people using distilled water. I use the Vallejo Thinner when I want thinner coats with Vallejo Air, and it works great.

For pigments, I apply them all the same, with a brush and sometimes water depending on effect desired.

Vallejo web site
I use plain old distilled water from the grocery store or pharmacy (chemist to you) at a couple buck a gallon to thin regular vallejo paints. Hey, it's what they recommend and tey should know!

Diluting Valleyo colors with alcohol leads to a huge mess! Your paint will become a sort of curdy substance impossible to use

As the other guys have told you, you can use distilled water, but your paint job will be a bit less resistant this way

Yes, DO NOT use alcohol or Tamiya thinner with Vallejo paints, unless you like cleaning out a goopy curdled mess from your airbrush.

Obviously the thinner that Vallejo makes is formulated, and will be the best product to use with their paints.

You can use distilled water, but you can run into the paint drying prematurely before it hits the kit. The Vallejo AB thinner has a retardant in it to help with that.

Of course if you just want to thin out the model color to make it easier to brush paint, then distilled water will do the trick just fine.

Yup all of the above!

I use their "Dilutant" looks like thin elmers'/PVA to thin Model Color + some "Tip Retardent" to slow drying out, esp. for brush painting.

I find use of distilled water will cause the colour to shift slightly when spraying, & the bond to plastic is not as strong.

If any of their Polyurethane Primers are in your base colour needs, use that, as it is tough as nails, & camo the normal ModelAir on top.

I also use a tiny bit of Medea a/b cleaner IF I need to thin ModelAir.