I think premelec has hit the nail on the head. You are wasting a lot of time fault finding with faulty test equipment. The very fact that everything is fine without the MM in circuit tells you that!
Cheap ready made test leads are often to blame. They are often crimped onto the wire insulation with very little copper making contact with the probes/clips. Also, the leads that come with a MM that has a 10A current range often cannot cope with 10A or will have such a high resistance that there is so much volt-drop that your circuit looks like it has a large resistance. The fuse is there to prevent fire. If it has blown, there is a high chance that the MM sense resistor has also blown, especially if the fault was due to placing the meter across a voltage whilst in current measuring mode.
As for your quest to size the battery for a robot, it sounds as if you already have a good feel for the motor current. Measuring will only give a guidline in the same way that the component specs will. The method I use is to fit a battery-pack (roughly spec'd from "feel"
and simply see how long it runs under simulated load. Erratic current demand affects batteries in peculiar ways and is very hard to predict even if you do have accurate current measurements for your motors.
I notice you are using Pb type. Have you considered how deep you are going to discharge them? I have found that anything deeper than about 25% reduces their life considerably. They are great for standby use but for deep cycle use I still find NiCd the best. NiMh are OK but still a little expensive and not as forgiving as NiCd. Pb is cheap though, even if rather heavy for comparable capacity.
