My data has two variables: treatment (untreated vs treated) and exon number (control, 1 and 9), each with 3 replicates. So 18 samples total but 6 unique groups (untreated-ctrl, treated-ctrl, untreated-1, treated-1, untreated-9, treated-9).

I started by coding everything into one design matrix, then making contrasts for all the comparisons I care about

# No intercept, all data
design <- model.matrix( ~0 + group)
contr.matrix <- makeContrasts(
  Ex1Unt = untreated_1 - untreated-ctrl, 
  Ex9Unt = untreated_9 - untreated-ctrl, 
  Ex1Treat = treated_1 - treated_ctrl, 
  Ex9Treat = treated_9 - treated_ctrl, 
  Ex1 = treated_1 - untreated_1,
  Ex9 = treated_9 - untreated_9,
  levels = colnames(design))

v <- voom(dge, design, plot=TRUE)
fit <- lmFit(v, design)
fit <- contrasts.fit(fit.voom, contrasts=contr.matrix)
fit<- eBayes(fit.voom, trend=TRUE)
summary(decideTests(fit))

       Ex1Unt Ex9Unt ...
Down     2099    968     ...
NotSig  12434  14619     ...
Up       1396    342    ...

My understanding is that since my design matrix is orthogonal, the intercept vs no-intercept approach should give me the same output. So I tried the analysis with an intercept but it gave me very different numbers.

This time I subsetted the data to just the untreated samples (untreated_ctrl, untreated_1, untreated_9)

# With intercept, just Untreated data
design_meanref <- model.matrix(~group)
v <- voom(dge, design_meanref, plot=TRUE)
fit <- lmFit(v,design_meanref)
fit <- eBayes(fit)
summary(decideTests(fit))

       (Intercept) groupingEx1 groupingEx9
Down           493        2473        1311
NotSig        1695       11336       12958
Up           13041        1420         960

My hunch was that this is different because this was just a subset, though I'm not sure why that would be the case. But I tried the first approach again, this time with the same subset.

# No intercept, just untreated data
         Ex1   Ex9
Down    2487  1333
NotSig 11319 12941
Up      1423   955

So two questions here:
1) Why were the numbers so different in my first and second analyses? Why did including all the data change the expression analysis so much even though I was looking at distinct groups in my comparisons? Which is more accurate?
2) Why did my third approach give me very close, but not the exact same result as the second?

Thank you!

Subsetting the data will always change the results, as it would do for any linear model or anova analysis, not just for limma. We always recommend that you analyse the whole experiment together.

It's hard to comment otherwise because you the code you show seems incomplete. The first code chunk (No intercept, all data) would not run unless you set the columns of design to be the same as the levels of group. I suspect you've omitted a line of code. The second code chunk (With intercept, just Untreated data) could not give the output shown because Ex1 and Ex2 are not levels of group. The third code chunk (No intercept, just untreated data) doesn't have any code so it isn't clear what you've done exactly.