Is it common for zh/j, ch/q, and sh/x to be merged?
As you probably know it is very common to mix up zh、ch、sh and z、c、s.

Here they say that in many dialects zh、ch、sh can't be distinguished from j、q、x.
But they don't mention which dialects.

On this page somebody says that Cantonese people can't distinguish the 3 groups of consonants in j,q,x,zh,ch,sh,z,c,s

On the same page they also write that the consonants jqx are pronounced as /zi/,/ci/,/si/ in Nanijng.

In the Chaoyang dialect there also seem to be such mix-ups. But not for all words. For example x、sh are read as j、q、zh、ch, for example in 喜(～鹊)、跹(翩～起舞)、械(机～化)、摔(～跟头)、尚(和～)、殊(特～)、匙(钥～)、始(开～)

j、q、x and z、c、s are often mixed-up. For example in: 俗(风～)、宿(～舍、住～)、狙(～击)、沮(～丧)、咀(～嚼)、俊(长得～)、峻(险～)、竣(～工)、骏(～马)、(～包)等。