I can't answer all of your questions, but this is what I know for sure.
Adding a random effect to clmm has much the same impact as adding a random term to a logistic regression model. Take a simple case with ordinal response $Y$, continuous covariate $X$ and grouping variable $G$. Write $p_k=Prob(Y \leq k)$, for some level of the ordinal response. The model posits that for subject $i$ in group $j$, $$\log \frac{p_k}{1-p_k}=\theta_k - \beta x_{ij} - g_j$$
Takeaways:
- We are modeling a chain of logistic regressions where the binary response is: "less than or equal to a certain level" vs "greater than that level".
- the $\theta$ parameter depends only on the ordinal response. It's like an intercept term, although there is one of these for each level.
- $\beta$ is the slope parameter for the fixed effect $x$, and it does not depend on $k$. The impact of $x$ applies proportionally to all levels of the response.
- $g_j$ is an unseen, grouping effect. clmm will estimate the value of $g_j$ and the variance parameter of its (assumed) normal distribution.
- The minus signs are a convention of clm and clmm. In interpreting the parameters, negative values imply a greater probability for low ranking responses and positive values imply that the odds favour higher levels of the response. I do not believe that this convention is universal. SPSS, I'm looking at you.
In this simple model, the random effect increases or decreases the odds ratio by a factor of $e^{-g_j}$. This factor is independent of the level itself. Say, if $e^{-g_j}=0.5$ in some group, then all odds ratios are cut by a half for members of that group. This assumption is the weak link of the proportional odds model.
You could include random slopes as well, if you believe there is an interaction between the group and covariate $x$.
You can extract the random effects that the model estimates using method ranef, as in ranef(clmm.complete).
Laplace approximation is a method of numerical integration. Random effects models involve an iterative procedure to estimate the random effects and the process parameters, the EM algorithm. I believe that's where the Laplace approximation comes in, since during the estimation phase, you have to integrate over the unknown parameters. However, I don't know the details of the algorithm employed by clmm. Perhaps someone else can give a better answer.
Be aware that clmm requires an optimization to produce estimates. As part of the summary function, you get some information about whether or not the optimization succeeded. The gradient should be near zero and the Hessian index should be less than 10000.
Regarding the specifics of your model, I don't quite understand the difference between variable SV and variable SVid. In any case, adding (1|SVid) basically changes the odds ratio for each group indexed by SVid. The model has to estimate one parameter for each SV group, a variance and possibly some covariance terms with other parameter estimates. Your description of the study suggests a lot of groups and not much data in each, so that could lead to problems with instability and convergence of the model.
I'm not sure what you mean by "grouping errors around individuals".
What you are terming 'main effects' are not really that in the usual sense. You cannot interpret lower-order effects (e.g., "main" effects) without simultaneously taking into account any higher-order (e.g. interaction) effects that are constructed of terms from lower-order effects. My Regression Modeling Strategies course notes have a detailed example of interpretation for a simple example where age has a linear effect and interacts with sex. It shows you how to do the composite "chunk" tests that are meaningful and are independent of how the variable are coded. For example, the age effect is the combined age and age x sex effects, which tests whether age is associated with Y for either sex. The sex effect is the chunk test for sex + age x sex interaction and tests whether there is a difference between the sexes at any age.
To get specific meaningful estimates, you form contrasts, e.g. difference between male and female at the median age.
Don't use statistical significance to choose a model. Stick with pre-specification, with an exception being this: if you do a chunk test of all interaction effects combined and the multiple degree of freedom test for all interactions yields p=0.4, you can fairly safely drop all the interaction terms. Some statisticians use AIC in making this decision.
The R rms package anova and summary functions do all this automatically, as shown in the detailed case studies in my notes. For more resources see http://fharrell.com/links.
Best Answer
First, as AlexK pointed out, this flipping of signs can happen because of the nature of multiple regressions. The coefficient in a multiple regression gives you the marginal effect of that variable with all other terms in the model held constant.
Let's look at interaction effects, but in a highly simplified 2x2 design. The principle can of course be extended to continuous variables, it is just a lot easier to think about it this way. Imagine your data set looked like this, with the means in the cells and the marginal means around it.
As you can see, both GDP and Corruption have a positive main effect, because in both "High" groups, means are higher. The model would be:
y = b0 + b1*GDP + b2*Corr (with both IV either 0 or 1).
Estimating coefficients would lead to both b1 and b2 being positive, due to the high mean when both are 1. The model has no chance but to estimate coefficients in a way that fit the data best, and with this model, it will estimate two positive coefficients.
Let's include the interaction term (GDP*Corr), which will actually be 0 in all but the first equation.
Now the model has a chance to account for the "diagonal" effect of both IVs being "High" together. The high mean of 20 can now be accounted for by b3. The model can estimate marginal effects holding the interaction effect constant and show that the remaining (=marginal) effects reflected by coefficients b1 and b2 are actually negative.
I had to add: The interpretation of something like this can become tricky very quickly (imagine three-way interactions). If you have a significant interaction, be very careful how you interpret it and any remaining main effects!