Isn't it true that denaturing only breaks weak bonds (ie- H bonds, NOT covalent ones)?
"Denaturing agents" break only weak bonds, not covalent bonds. Some might consider mercaptoethanol to be an aid in denaturation and so classify it as a denaturing agent. In that exceptional case, a covalent bond would be broken. But never a peptide bond.

Can a strong salt denature proteins by disrupting ionic bonds? As in, will the salt ions compete with side chains to form ionic bonds?
High salt can denature proteins, but more often it counteracts repulsive ionic charges that keep proteins from associating illegitimately with each other. Thus a high concentration of ammonium sulfate is often used in protein purification: at a given concentration (several molar), certain proteins will precipiate out of solution, while others will not, so you can collect the precipitate, return it to a low salt condition, and thus resolubilize the native (never unfolded) protein.