The point of roman numeral analysis is to represent what certain chords "do" in the key, or how they functional (and it's often called "functional analysis'). If the music wasn't written to be functional (like a soundscape, early polyphony, or some modern music), it's not very helpful to analyze with functionality in mind. Some kinds of music are better analyzed using letter names, if the chord progressions are generally non-functional.
If you run into chords built on non-scale degrees, you can notate them using bX or #X. These are generally uncommon in "common practice" Western music, with a couple of exceptions - like the Neapolitan 6 or N6, built on the bII. The roman numerals are only modified if they are unexpected in the key - i.e. if they're built on something other than a scale degree. In the key of A major, and F chord would be a bVI, because F# is expected, but in a minor an F chord would just be a VI, because F(natural) is expected.
If you're just analyzing into roman numerals to aid in transposition, you can stop here - that's all you need. If you want to understand how and why the composer picked a particular chord, you can go much deeper into the composer's thoughts.
Be aware that roman numerals go by the spelled name, not any enharmonic equivalents, unless you have reason to believe the publisher didn't notate the music with the analysis in mind. So, for example, an E# chord in the key of A would be a #V, not a bVI. Usually composers are aware of how their chords function and publishers also analyze music before typesetting it.
A final thing to bear in mind is that x7 chords that don't belong in the key are often Secondary Dominants, or chords that are borrowed from another key. These are usually notated as either V7/x or vii/x where x is the key their borrowed from relative to the root. The most common is the V7/V, or "five-seven of five", which is not a borg designation but rather the five chord borrowed from the key of the five chord. In the key of A this would be a B7 chord. So before you start labeling II7s in a major key, realize they're probably V7/V.
The borrowed Secondary Dominant chords "stand in" for the chord they're borrowed from, and often, but not always, set up and resolve to that chord.
Here's your example using a plausible borrowed chord:
| A | A7 | D | Dm7 G7 |
| I | I7 | IV| ii7/III V/III|
| A F#7 | F E7 | A D | A E7 |
| I VI7 | bVI V7 | I IV| I V7 |
or N/V V7
If you don't see any way for a chord to be borrowed, look for non-chord tones. Sometimes the chord can be disguised by the inclusion of a note that is outside of the chord. You'll see notes leading to and from the suspect pitch in a melody or counter-melody.
Sometimes there is more than one way to analyze a particular chord, in that case, look at how it resolves and pick the analysis that best explains the resolution.
