My hypothesis is that all these examples of "suspect algebra" are really examples of "imitative algebra"¹.

Much learning is imitation, something that we are basically "hard-wired" for, and therefore lying largely beyond of the constraints of deliberative/logical reasoning. It takes some training to turn off this tendency to "learn by imitation" (in which reasoning plays no role) in contexts, such as learning math, where it is inappropriate.

My point is: don't be alarmed; IMO, what you're seeing is perfectly normal. "It's just a phase," as they say.

My advice would be, first: don't have a cow over such doozies. (It'd be like despairing over the incomprehensibility of an infant's babbling.) (BTW, I suspect that overreacting to such errors may be the genesis of, or at least contribute significantly to, life-long "math phobia".)

Second: use these mistake as teaching opportunities. For example, when you come across something like

$$\frac{1}{a+b} \mathrel{\text{“=”}} \frac{1}{a} + \frac{1}{b}$$

ask the student to check the equality by replacing $a$ and $b$ with some actual numbers. Learning how to check one's derivations is a crucial, and extremely general, skill, far more important than any one algebraic "rule", and the sooner such "derivation self-checking" becomes second-nature, the better.

_{¹ Infants and young toddlers babble. The hypothesis that babbling is an imitation of talking is supported by the fact that children (whether hearing or not) of parents who use sign language will display "manual babbling" at the stage when "vocal babbling" normally occurs. Less common than babbling, but in a similar vein, some pre-school children will display "mock reading".}