Something fancier than a univariable or multiple regression model is needed when there is a very clear, very complicated non-linear relationship between the endpoint and covariates that cannot be addressed using a routine link function or transformed covariate. If your paneled scatter plots show clouds of data points with suggestive trends then anything fancier could be considered overfitting.
Based on your labeling of what is the dependent and independent variables, your model will tell you if any DV's (high or low) are more common in words with certain properties. Your description had this reversed, suggesting reversal of dependent and independent variables.
Yes, with a large enough sample some of your predictors are bound to be statistically significant, and that is a good thing. In fact, it is a great thing. Don't be disappointed with small p-values. However, just because you have the power to detect an effect does not imply it is a meaningful effect to discuss and report.
As decaf and Eoin have suggested, you can use LASSO and FDR methods among others to weed out independent variables. My preference is to use the p-value as a continuous index for the weight of the evidence by running univariable models and sorting the predictors by ascending p-value, as well as by the magnitude of the estimated effect size. Those predictors with the smallest p-values do the best job explaining your endpoint for your particular data set and stand the best chance of re-demonstrating their effects in a repeated experiment. If this "short list" is still a bit too long you can choose to highlight only the top, say, $k$ independent variables on the list. If we think in terms of Neyman-Pearson error rates instead of evidential p-values, we can talk in terms of per-comparison error rates so that no adjustments to the p-values are necessary, acknowledging that family-wise error rates may very well be higher. The key is to interpret the results as tentative evidence, not irrefutable proof. Those independent variables with the largest effect sizes are the most interesting to discuss and report, even if the p-value does not reach a conventional cutoff like 0.05.
The independent variables that were promising in univariable models can be explored together in a multivariable model to see if the conditional effects persist in smaller subgroups. I suggest creating paneled and grouped plots to explore possible interaction effects. However, this can become very nebulous very quickly. For that reason I prefer to keep emphasis on the univariable analyses that describe the higher-level conditional effect for each independent variable. This may seem elementary, but it may be naive to slice and dice the data by a dozen or more independent variables simultaneously and discuss a miniscule effect in an obscure subgroup especially when the results are tentative evidence and not irrefutable proof. It may be worthwhile to define a subpopulation and fit a simpler model to the data available on this subpopulation, rather than trying to interpret a multivariable regression model. Such a stratified model may provide a better fit to the data and the reduced sample size will produce more conservative inference.
No matter what modeling technique you choose the only true validation is to repeat the experiment and see if the same model does well at explaining the dependent variable. If the experiment can be replicated many times then what was tentative evidence, taken together, becomes irrefutable evidence.