Google-sheets – Want to use a cell to get a limit number

Indeed, one cannot put things like "A6" inside the query string because then it would just a part of a string, not a cell reference. Instead, concatenate the value of the cell to the string:

=JOIN(", ", UNIQUE(QUERY(A2:F4, "select E where day(A) ="&day(A6))))

You don't need transpose, because join works equally well with one-dimensional arrays of either direction: vertical or horizontal.

That said, I find a solution using filter slightly more readable:

=join(", ", unique(filter(E$2:E$4, A$2:A$4 = A6)))

The arrayformula(query(...)) combination is not supported; there is no concept of a query processing an array of arrays or executing an array of query strings.

You have two options: (a) repeat query on every row; (b) use vlookup to retrieve columns of data, as explained below. For example:

=arrayformula(vlookup(E3:E, {A3:A, J3:J}, 2, false))

takes one element of E3:E at a time, finds this element in column A, and retrieves the corresponding element of column J. With this approach you would need six separate vlookups to get the columns J,I,H,G,F,E, but you won't need a separate command for each row.

A complication is that vlookup accepts only one search key, and you want to search by 4 parameters,

A='"&E3&"' and 
B='"&B3&"' and 
C="&C3&" and 
D="&D3&

This can be worked around by concatenating these into one search key: you can have a column like Z,

=arrayformula(E3:E & "|" & B3:B & "|" & C3:C & "|" & D3:D)

which concatenates four search parameters into one pipe-delimited search key. Do the same for the table to be searched, and then compare these keys using vlookup.