Could a solid object made entirely of antimatter safely "touch" ordinary matter without annihilating each other because of EM force? Most is said about the anti-particles annihilating their ordinary particles counterparts. 
When antimatter forms anti-atoms, it is said they behave like ordinary matter. 
Since atoms are more stable them their free particles, could it be possible for interaction between antimatter atoms and ordinary matter atoms be a safe thing? 
Experiments use mostly "free" positrons and anti-protons. We know about how "free" electrons and protons interact with ordinary matter, and they can do a lot of "damage". But when they are together, in atoms, they become much more stable and form bonds, lattices, molecules and so. Atoms in ordinary matter does not "touch" each other particles because of the EM force. They interact yes, but a direct collision is rare thing.
 A: It depends on how many antiparticles would react with ordinary particles. The few that do due to cosmic particles and particle accelerators seem safe enough.
In empty space with 1kg ball of iron and 1kg of anti-iron it would be difficult to make them react completely, for the same reason it is difficult to make all plutonium in a nuclear fission bomb react completely - the initial release of energy is so great that it in most cases hammers out the interacting parts away from each other and stops further reaction. Such failed explosion (which still may be very strong) is called a fizzle. It takes either extreme velocities or precision shaped charges to make most of the plutonium react in the bomb. Presumably the same would be true for matter-antimatter reaction. These things wouldn't annihilate completely just because they get close to each other, it would take special conditions for that to happen.
A: If solid objects of matter and antimatter touch, annihilation is inevitable. You would not want to hold a clump of antimatter in your hand. The amount of energy released (~25 MT) would be comparable to an asteroid or comet hitting the Earth.
Some food for thought here.
A: Treating this like a Fermi problem: Suppose we have two spheres of (terrene) iron and we bring them slowly into contact.  The first significant source of a contact force will be from electron-electron interactions, and since electrons have like charges, we expect the contact force to start out repulsive and only get more strongly so as Pauli exclusion also comes into play.
If we replace one sphere with anti-iron, however, now those first interactions are between electrons in the iron sphere and positrons in the anti-iron sphere, so the charges are opposite and we expect the contact force to be attractive initially.
