FBS is used as relatively inexpensive source of nutrients that support cell growth.
One of the reasons for using fetal over the much more abundant and cheaper adult bovine serum is the low level of
antibody and complement in the fetal circulation. Therefore, your serum is less likely to kill your cells. Nonetheless, serum is often inactivated by keeping it at 56°C for 30 min, which further reduces complement activity.
Serum contains many different factors that aid cell growth and there is no complete understanding of which factors are important for which cells. A quick list off the top my head: lipoproteins (source of cholesterol, vitamin E, lipids), transferrin (source of iron), other metal carrier proteins, retinol binding protein (vitamin A), cytokines (long list), insulin, other hormones, and on and on. It is also worth remembering that cells do not live in a salt solution environment, rather they are surrounded by high concentrations of protein and other cells. The abundance of albumin in FBS helps to increase the overall protein concentration in your medium.
It is also important to mention that serum contains factors that may actually inhibit the growth of your cells. Most likely, each batch of serum, pooled from many animals, contains a mixture of factors that help your cells grow and factors that inhibit their growth. That is why it is important to test batches of serum for which one is best for your cells in whatever assay(s) your lab is using.
Serum is available for other sources: human, horse, pig, mouse etc. They are all different and have different effects on cells.
As for your question regarding the health of your cells, it is hard to tell without knowing what cells your growing and seeing them. Some cells are very picky others can handle some abuse. In general, you have to split them as they become confluent or the medium becomes acidic. Some cells don't like to be split too much (1:3) others can handle a 1:10 split or more. If your cells overgrow and then you split them hard, you could end up with a very unhappy culture that takes time to recover.
Whoever is training you should fill you in on the basics of your cells and you can also look up this information from ATCC (if they sell your cell type) or look at the original publications describing your cell line.