But how do I find out if a word is not on a line

Just use the negated version of the regex match e.g

shell> perl -le 'print "yep" if "foo bar" !~ /baz/'
yep
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The second thing i would like to know is how do I remove the first char from a string in an array element?

Using substr is a pretty sensible approach e.g

shell> perl -le '$_="xfoo"; print; substr($_,0,1)=""; print'
xfoo
foo
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Or you could use a simple s/// e.g

shell> perl -le '$_="xfoo"; print; s/^.//; print'
xfoo
foo
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The third thing is i want to remove all chars up to the last ',' in an array element.

You could use a naive regex for this which would just match a comma and then everything that wasn't a comma to the end of the string

shell> perl -le '$_="foo,bar,baz"; print; s/,[^,]+$//; print'
foo,bar,baz
foo,bar
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But if you're parsing comma-delimited fields then the likes of Text::xSV is a much saner approach.

The above answers can be applied to an array element just as easily a simple scalar variable as they're both scalar values.