I have a DV (NPS) on a "1 = not likely at all" - "10 =extremely likely" scale.
My two IVs are on an ordinal scale (1-3 with N/A option).
<1,2,3,NA> is not strictly ordinal, though in some particular situations it might be reasonable to treat NA as if it were at one end, or as missing in which case you could have it as ordinal. In some other situations it might be reasonable to split it into two variables (NA/not-NA and ordinal 1-3 given not-NA). In still other situations it might be reasonable to treat it as nominal.
I see many different discussions online if NPS can be used as ordinal or interval scale and am now quite confused how I should treat it. Would be grateful if someone could shed some light.
When there are many ordinal categories it's not unusual to treat it in an interval-like manner but fit low-order polynomials (so that relationships are smooth but not necessarily linear). An alternative might be to fit smooth functions such as natural splines. This may be quite reasonable with large sample sizes.
(Polynomial contrasts are the default in R if you put an ordered factor in as an IV in regression, for example)
A second possibility is to fit strictly ordinal categories through approaches like ordinal logistic regression/proportional odds (there are a number of other possibilities suitable for ordinal data). If you treat as unordered you might do multinomial logit.
My DV is not normally distributed (skewed to the left as majority of replies are in the >7 region)
It's categorical, so it can't be normal, but marginal normality isn't an assumption of regression anyway. The main question is more whether the conditional distributions are close enough to normal that inference won't be badly affected at the sample size you have. But even if normal theory inference is not tenable there are other things that might be done.