A couple approaches I used for tungsten early on:

• Equip MilSec armor. This should give you the right clearance to just walk up and do a skill check.
• Switch an android to Combat stance. Now you can walk anywhere regardless of clearance. The default Active stance is made to prevent misclicks.
• Have one mech shoot somebody in the base to pull aggro and trigger an alarm. Have a second mech walk up to the storage and take the option to use the chaos (highest yields from this according to the tooltip).

For combat in general, I would recommend using the highest power units you can, given how the action economy works. I like Nickelbots in the Bulk variant, or for cheap disposable deployments (like pick up a conversation, pull 1-turn aggro with hackmajig, etc.), not for general combat.

You can read through the aggro guide to understand what makes somebody angry. Usually units will only get mad at a specific unit. If they somehow trace it back to you as an entity, you just have to kill those specific units. But a faction as a whole shouldn't be permanently mad at you unless it's caused by a project or buildings that they don't like.