So the minutemen doesn't get drill I, it gets drill ii if it upgrades from a unit with drill I.
Yes, Minutemen get Drill I for free. No, they won't get Drill II for free if you build/buy them in the city that has Alahambra.
In G&K if you upgrade from a longsword that got Drill I by using a promotion instead of also receiving it free from having Alahambra, you've effectively wasted that promotion. Especially if you built/bought that longsword in a city with a barracks, but without Alahambra, on the turn before unlocking minutemen, don't have promotion saving enabled and gave it Drill I with the 15 exp gained from the barracks, because not for Drill II you'll need an additional 15 exp or 30 total. However, with Alahambra that longsword would have started with Drill I for free so the first 15 exp could have been used for Drill II.
City with Barracks & Armory
Longsword - Drill I & Drill II
Minuteman - Drill I (always free), Drill II & Drill III
City with Alahambra, Barracks & Armory
Longsword - Drill I (free from Alahambra), Drill II & Drill III
Minuteman - Drill I (always free), Drill II & Drill III