How to split field 1 into distinct rows but keep field 2 copied for each new row created

Well, the first field is $1, NF holds the number of fields on the line, we can access the fields with $i where i is a variable, and loops work almost like in C. So:

$ awk '{for (i = 2; i <= NF; i++) printf "%s\t%s\n", $1, $i} ' < blah
1       the
1       mouse
...

(This doesn't differentiate between space and tab as field separator.)

With GNU sed:

sed -E 's/^((\S+\s+)\S+)\s+/&\n\2/;P;D'

It's just uglier with POSIX sed syntax:

s='[[:space:]]\{1,\}' S='[^[:space:]]\{1,\}'
sed "s/^\(\($S$s\)$S\)$s/&\\
\2/;P;D"

another awk one:

~$>echo '1   the mouse is dead
2   hit the wall
3   winter lasts forever
' | awk 'BEGIN { RS="[[:space:]]+"; } /^[[:digit:]]+$/ {line=$1; next}; { print line "\t" $1; }'
1   the
1   mouse
1   is
1   dead
2   hit
2   the
2   wall
3   winter
3   lasts
3   forever

and laid out slightly better..

# split all parts into single word records.
BEGIN { RS="[[:space:]]+"; } 

# if the record is a number the save
/^[[:digit:]]+$/ { line=$1; next }; 
# else use last saved line number and this record to format output.
{ print line "\t" $1; }