1. You can estimate mean and SD of each component of fasting and random lipid profile and compare with t-test.
2. You can categorize the fasting and random lipid profile and compare with Chi-square test. Categorization may be of individual component basis or overall.
3. You can find out the correlation of each component of fasting and random lipid profile by Karl Pearson's Correlation Co-efficient.
As you want to compare fasting and random lipid profiles of the same persons, you need to do paired sample tests. And as the lipid profiles tend to show a skewed distribution you have to do Wilcoxon signed-rank test and express the values in median (interquartile range).
However, you can also categorize the lipid levels and do a McNemar test (as these observations are paired).
Regarding mixed model analysis,
You only do mixed model analysis when you want to compare two paired observations and want to plug in covariates and factors. If you are considering factors like participants' food habits, items, time of measurements to see their effects on the difference between fasting and random measurements of lipid profile, you can think of going for mixed model analysis (GLM).