...Is this a technically "correct" progression?
There is no such thing as a correct progression. The only place you have to deal with that is when a teacher corrects a 4 part chorale harmonization :-) My point is that technically correct only applies to the extent that the style of the music has such technicalities.
Your chords fit into
A major. Assuming you repeat this progression the changes
D F#m hit the subdominant region then
Esus4 E hits the dominant region, leading finally to the tonic
A at the repeat. Subdominant to dominant to tonic is the model of functional harmony. Also, you are aware that you are using the common
I V IV V harmonic template.
You are on solid harmonic ground. Rhythm and melody should become the next obvious points of concern.
...I know Esus4 would typically resolve to E, but it sounds good resolving to D.
In styles that use jazz symbols
sus4 chord very often don't behave like 'classical' counterpoint suspensions. Technically in that counterpoint style the suspended tone would be the
A and it would resolve to
G#. I'm not sure of your exact chord voicings, but I imagine you are holding an
A tone over chord
D and I think it then continues into the
F#m. You could think of that more as a pedal tone than a suspension. If the
A continues to be held into the next
Esus4 and then resolves to the
G# on the final
E chord, then the ending provides the traditional suspension resolution.
...How big of a difference do inversions make?
The the classical style it makes a big difference as the bass moves more melodically and voice leading rules are applied. IMO, In rock style, inversions are often used simply for the sonority and there isn't any concern for voice leading. In other words: in classical style 2nd inversion is dissonant and technically requires resolution, but in rock style 2nd inversion has a sort of 'rumbling' sound that is desirable and does not require resolution. I'm thinking of power chords in 2nd inversion. But you could also exploit the sonority of inversions in other styles too.
One important thing to keep in mind is the difference between chord inversion on instruments like piano and guitar and whatever might be played by the bass.
Pianists and guitarist often speak of chord voicings. In isolation these voicings are really chord inversions. So a drop voicing could be a 2nd inversion chord. If a bass is playing at the same time and playing chord roots then technically the harmony is in root position.
Chord voicings on piano and guitar will most likely follow smooth voice leading, but in terms of inversions and the actual harmony be sure to look at what the real bass part is doing.
...Do the inversions have any effect on the progression in terms of the key or mode of the song?
None that I can think of.
In classical style the contrapuntal handling of the inversions really will clarify the key.
In rock style inversion often are about sonority or you might use the word timbre. I think in terms of acoustics the various inversions reinforce different overtones. Overtones are responsible for timbre.
It's hard to comment further without more musical detail. Any concern about composition technique will depend on those details and the style of the song.