Now why is it generally ok to cross out CO on both sites?
Every chemical reaction tells you about the amount of reactant used up, and the amount of product made. To make things easier, we will consider a situation where exactly one mole reacts, e.g. a stoichiometric coefficient of "1" in front of a reactant means we use up one mole, and a stoichiometric coefficient of "1" in front of a product means we make one mole.
To understand why we can cancel CO on both sides, imagine we have a thousand moles of all species present (reactants and products). Each reaction will change the amounts in our stockpile. Because there is so much, we will never run out.
The first reaction adds a mole of CO to our stockpile. The second reaction uses up a mole of CO in our stockpile. The net effect is no change in CO. It does not matter whether reaction (1) happens first, second, or at the same time as reaction (2).
Or do I always have to imagine with such manipulations that they are pure mathematical steps to the actual reaction of interest without the various reaction paths actually reacting in such ways?
If the reactions are real, substracting or adding reactions is real, as well. When you add them in a one-to-one fashion, there is usually a physical reason. For example, if you start out with no CO at all, the second reaction can only use up as much CO (or less) as the first reaction produces.
Why can we add/substract/cross out chemical equations for Hess law?
When you substract a reaction, you are saying that it runs in the reverse direction. In this case, your stockpile of products will diminish, and your stockpile of reactants will increase. Again, this might actual happen, or it might be artificial.
For Hess' law, it does not matter. The quantity you calculate with Hess' law, enthalpy, is a state function. Any path from A to B will result in the same enthalpy change, no matter whether real or fictitious. If you can measure the reaction from B to A, for example, you can also give the enthalpy for the reaction from A to B, even if you can't do the experiment.
[OP in comments] "If the reactions are real, substracting or adding reactions is real, as well." - That is a little unclear to me, although I can have two real reactions, but if I add them together, it can also happen that instead of the products merely being additive, another products are formed because, for example, a reactant from reaction a now suddenly prefers to bond with a reactant from reaction b. A+B->C and D+E->F, instead of A+B+D+E->C+F it is A+B+D+E->I+J, where A+D->I and B+E->J. That shouldn't be simply additive anymore because it doesn't produce the same products as the 1. and 2. reactions alone.
That is true in general. If I don't mix reactant, I don't get a reaction. If I add a third thing (maybe an acid or a base, for example), I might get a different product. Hess's law just says that once you describe a certain set of reactions with some reaction equations, the enthalpies add up in the same way that these component reactions do.