At first instance, why do we do pathway analysis and gene enrichment
Sorry, please do your own background reading in order to understand this. Go to a search engine, type in keywords
analysis, and then start to read.
Have a set of genes which are upregulated and down regulated between
wild type and mutant, how to get enrichment score for upregulated
genes and down regulated genes?
Perform the enrichment separately, using the direction of fold-change to determine up- and down-regulation
How to identify which pathways are enriched in the wild type and
Different ways to do this. This could be the same as the answer that I gave in the previous point, or you could define a threshold Z-score for 'expressed' 'not expressed' (using the entire unfiltered dataset), and perform the enrichment and / or pathway analysis separately on those genes passing the threshold in wild type and, then, mutant.
How to identify which pathways are enriched in upregulated or
Perform the pathway analysis separately, using the direction of fold-change to determine up- and down-regulation
Some resources to get you started: