Do sit down - this is quite a long post. In answer to your question, this has some features that sound like Beethoven but a lot that doesn't, particularly regarding the use of key areas. As you have stated your intention to imitate the style of Beethoven I shall be basing much of my commentary on comparing your sketch to what he does in his chamber works. I have attached a pdf so that you can see more clearly what I am referring to and a version I'm made with some suggested improvements. I am making the assumption that since Beethoven would have done so, you are writing this movement in sonata form.
Bars 1-4: I see how your first theme quite clearly takes after the D maj quartet. The device of stating the motif and repeating it with variation is a good way to start. However, what Beethoven adds is a decoration, a little turn which gives the motif more energy and makes it more distinctive than simply a dotted crotchet-quaver-crotchet rhythm. If you look at the first movement of the Op.1 no 3 piano trio, he does a similar thing in the initial motif, also played unison by the four parts. It doesn't really affect the motif harmonically, so it can be maintained wherever it appears subsequently, but it does give it a particular individuality that makes it recognisable when it occurs.
A few problems with the exposition of the subject here. Firstly, the motif astonishingly moves to the subdominant in the first statement! This sounds 'wrong' generally, but especially in a classical-era piece. The first statement of a subject like this must either remain in the tonic (as happens in 95% of cases) or move to the dominant (the opening of Brahms 2 gets to a dominant pedal in the second bar, but only reinforce the sense of the tonic when it returns). On the second statement you could go to the subdominant/chord IV, but this is something Schubert, rather than Beethoven, is more likely to do. This is connected to the second problem, the voicing of the chords in measures 2 and 4. They are both inversions without the lowest note at the root, adding to the odd harmonic character here. It might be better to maintain the unison as it will heighten the novelty of harmony arriving later on. Again, listen to the opening of the Beethoven quartet - he saves harmony for the third phrase of the subject. Thirdly, the doubling of viola and 'cello in the first bar is sonically good, but forces the viola to move upwards for the chord. Better to have up the octave doubling the violin so it can move downwards with it.
Bars 5-8: Harmony here is way too modern to be the late classical. Mostly this is because you have parallel harmonies or harmonies that move to odd places - the chord sequence is something like Ic-II-VI-IVm7-V which Beethoven would never write, as it sounds awkward. Also, you finish the phrase on a bare fifth with no third which is again atypical.
Bars 9-13: Same odd harmony as the opening, although it doesn't sound quite so odd now the piece has got going. Bar 11-12 nice tonic-inverted dominant progression. Bar 13 is rather strange and, I think, superfluous. The viola establishes a decorated pedal, which this is a good way to introduce a new subject (although the F-E against the cello, which is clearly in B minor, sounds very odd). However the 'cello's arpeggio-like figure sounds like the new subject and therefore the entry of the violin in bar 14 with the actual subject makes this little episode sound out of place...
Bars 14- 25: ...but wait, it can't be the second subject, because it's still in the tonic! A nice little tune actually, but the harmony parts move rather awkwardly. Try singing the parts and you’ll see what I mean. Don’t forget stringed instruments can play chords. I really don’t like bars 21-23 as you’re trying to ‘modulate’ without actually using the accidentals in the new key, but rather through sevenths and ‘jazz’ chords which sound very awkward without the black notes. Highly un-Beethovenian.
Bars 26-33: Worse still, PARALLEL FOURTHS! Completely against the rules of harmony in Beethoven’s time, and bad part voice-leading practice anyway, unless you’re Hindemith or Bartok. In any case the chords are very strange ones considering the harmony will move back to D major yet again before the pause. (I assume the blank bars are ‘work in progress’).
Bars 34-36: Another rather awkward sequence of chords leads us back to...D major and an exact repetition of the subject. Thirty-five bars and we haven’t even reached the dominant yet. This is going to be a long exposition – you sure you haven’t been listening to Bruckner?
Learning by imitating Beethoven is a laudable aim, but you should study how he moves between chords and how material is used structurally. If you want to write in the classical style, it is imperative that you understand the function of harmony in this period, especially the tonic-dominant relationship. My advice to you now is to start again and rework this exposition with a clearer form. Try to avoid copying what I've done with it though. After that try and write the development and recap. But above all, listen to and play as much Beethoven, Haydn and Clementi as you can to study what they do and why.
M Chamberlain string trio.pdf