p-values are commonly misinterpreted as the probability of a type-I error, i.e. that the null hypothesis is true but is being rejected. However, the p-value is actually a random variable. A nice paper covering this can be found in [1].
p-hacking also seems to be an abundant problem, i.e. the replication crisis in psychology [2], and it's not necessarily done on purpose [3].
Regarding the threshold for statistical significance, there are arguments for abandoning the concept altogether [4].