I got the Horseradish, Thyme, Chaparral, and Eucalyptus Oil for lung infections. I am wondering now, how do I mix the Horseradish, Thyme, Chaparral and get an even distribution?

In this case equal parts by weight.  For example 1 ounce each of the powders.  You can put these in a plastic bag, create a big air pocket and shake them real well so they are blended thoroughly.

I would guess that the herbs don't have the same density and therefore one would drop to the bottom when mixing. For that matter, how do you properly mix any herbs together?

Same as above is an easy way.  But ratios of herbs can vary a lot depending on the formula.

Also, can you please remind me the dose of this I need to take?

If you are doing capsules I recommend 3 size 'O' capsules or 2 size 'OO' capsules 3 times daily before meals along with a capsule containing 8 to 10 drops of eucalyptus essential oil.

Also, what dose could I give to a 3 year old and a 5 year old (I know NOT to give them the Eucalyptus oil, but I believe you said the other three are OK for kids)?

I recommend two size 'O' capsules 3 times daily at least 20 minutes before meals of the powdered herb mix.  And don't forget plenty of water to help keep the mucous thinned out.