Can the expression "toys out of the pram" be applicable for describing that someone is having a bad day?
No, "throwing toys out of the pram" is used for a person who is behaving irrationally (in anger or frustration). Here the person is being compared to a baby who keeps throwing the toys out and the implication is that the person is being irrational.
For your question, to describe a person who is having a bad day, some say "Got up on the wrong side of the bed".
People also tend to say that they are "having a bad hair day", if they are not having a good day. I have heard some say "My computer is having a bad hair day", when it continuously hangs on them.
I found the following at The Phrase Finder in response to a question about the phrase throw the dolly out of the pram:
In other words, a temper tantrum.