Monograph from IFSCC on stability testing which I think you will find useful: https://ifscc.org/wp-content/uploads/2018/05/2-Fundamentals-of-Stability-Testing.pdf
To answer your first question, in addition to viscosity and thermal stability, I would add pH testing (essential for skin/ingredient compatibility and formulation stability), centrifuge testing (quick look at stability, cannot be relied upon, not always accurate representation of real life conditions), microscopic analysis (not essential but helpful for looking at droplet size and distribution), microbial stability/preservative efficacy testing (essential, self explanatory), cycling testing, sensory evaluation (user experience), photostability (light exposure, stability, I’d say important but not essential), packaging compatibility (asses product/packaging compatibility), and rheological testing (flow behaviour).
There are other types of testing too but it depends on what you’re formulating, what your end goals are etc. I’d encourage you to read the monograph at a minimum, it is a basic overview of stability, easy to read (even novice formulators can understand). There are also multiple chapters on both subjects in various textbooks, if you want to learn they would be your best avenue.
As for your second question, to conduct a simple spreadability test, you do not need complex equipment. The glass plate method is easy to set up. You place a known quantity of lotion, typically around 0.5-1.0g, between two glass plates. Then apply a specific weight on the top plate (e.g.. 500g), leave for a set time (e.g. 1min). Then you remove the weight, measure the diameter or area of the lotion spread (larger area = better spreadability). This test is best when comparing different iterations (or even batch to batch) especially when adjusting emollients, thickeners or emulsifiers. You could also use a texture analyser, if available, to measure the force needed to spread the product. This gives you numerical data, which is useful for tracking formulation improvements. Even sensory panels can give insight. Have multiple people apply a fixed amount of lotion to the skin and rate it on ease of spread, greasiness and after-feel but this is subjective. Plate testing is easy, quick and cheap.