Insert Text – How to Insert Text in Specific Lines of a File

In your linked question there is already good awk answer, just modify it a little bit by using printf instead of print to insert the content without newline:

awk '/First/ { printf $0; getline < "File1.txt" }1' infile.txt

Result:

Some Text here
FirstThis is text to be inserted into the File.
Second
Some Text here

You may want to add space or other delimeter after "First" with printf $0 " "; ...

If inserted file has many lines then:

awk '/First/{printf $0; while(getline line<"File1.txt"){print line};next}1' infile.txt

Result:

Some Text here
First[
    {
        "foo": "bar", 
        "baz": "biff",
        "data": [
            {
                "a": 1945619, 
                "b": [
                    {
                        "c": 512665, 
                        "d": "futz"
                    }
                ]
            }
        ]
    }
]
Second
Some Text here

Here's one that works in the simple case of only one level of commands within an \edit{...}, at maximum:

perl -00 -lpe 's,\\edit\{( (?: [^}\\]* | \\[a-z]+\{[^}]*\} )+ )\},$1,xg'

The middle part (?: [^}\\]* | \\[a-z]+\{[^}]*\} )+ has to alternatives: [^}\\]* matches any string with no closing brace or backslash (regular text); and \\[a-z]+\{[^}]*\} matches anything with backslash, lowercase letters, and then a matched pair of braces (like \url{whatever...}). The grouping (?:...)+ repeats those alternatives, and the outer parenthesis capture, so we can replace the match with just the part inside \edit{...}.

-00 tells Perl to handle the input one paragraph at time, with empty lines separating paragraphs. If you need to handle tags that span paragraphs, change that to -0 to handle the whole input in one go.

For your example, this seems to work, giving:

We Introduce a model for analyzing \emph{data} from various
experimental designs, such as paired or \url{http://www/}
longitudinal; as was done 1984 by NN \cite{mycitation} and by NNN
\cite{mycitation2}.

However, it (predictably) fails for an input with two levels of commands inside the \edit{...}:

Some \edit{\somecmd{\emph{nested} commands} here}.

Turns to:

Some \somecmd{\emph{nested} commands here}.

(the wrong closing brace is removed)

Actually handling balanced parenthesis is somewhat more tricky, it's discussed e.g. in this question on SO: Perl regular expression: match nested brackets.