Well, the mass is not conserved - your own cells are not made up of the exact same components today as they were a few years ago. Cells constantly produce waste via a number of processes, and take up (and release) a wide variety of substances. This is why we need to ingest food on a regular basis - it provides the energy for the chemical reactions and substrates for metabolism and division, repair, etc of the cells.The same applies to single-celled organisms - they metabolize nutrients to produce waste products and repair and replicate the cell. The division process is not equal either - the daughter cells are made up of 50% or less of the parent, then add mass by ingesting nutrients. the so-called immortality of single cell organisms (and some multicellular too) is a bit of a misnomer: there are two types of division here - asymmetric and symmetric. In the asymmetric, only the daughter cell is "refreshed", in the symmetric (also called binary fission ), both are "refreshed", but I am not sure that the exact mechanisms for this are known.
Free radicals are very short lived in real-life, they are highly reactive and are actively scavenged by a number of substances produced by most (I don't know about all) organisms - a common scavenger is (hydrogen) peroxide. These do not accumulate in the cells to any degree. Mutations in DNA induced by radiation and free-radical reactions can be repaired by cellular mechanisms in most instances, and those that do accumulate are not guaranteed to cause a deleterious effect or to be in a region that codes for a protein. There is also redundancy in coding for most amino-acids, and usually some flexibility in which amino-acid can occupy which position in the protein (look at the charge properties and structures of the amino acids). In many instances there is also redundancy for proteins - several proteins are coded that can perform the same function. Those mutations that are deleterious might not have a significant effect on the organism - perhaps it takes a little longer to process a metabolite.
It should also be noted that the DNA replication process is not perfect either, it's very very good, but each replication cycle will result in a few base pair changes over a genome - the same as might result from radiation etc.
Serious damage to the cell results in death of the cell - ever had a sunburn? The dry flaky skin that sloughs off is the cells that didn't manage to repair the damage - the ones that did survive might have accumulated mutations, some of which can lead to cancers (e.g. melanoma). In the case of unicellular organisms, the individual cell will die, or potentially be selected against in the environment (if the mutation causes a significant deleterious effect).
You must remember that for any group of organisms, there is a range (bell curve/normal distribution) of conditions in which they will be fit, and some will be "more fit" than others -better suited to the conditions in which they grow, and so will reproduce more/faster etc and in theory dominate the population to become "more fit" over time (narrower bell curve). However this isn't the full story - each generation of those cells will contain mutations that are capable of being selected for and against, shifting the bell curve peak to the left or right. So this accumulation of mutations is actually a good thing for a species overall, as it allows adaptation to new conditions over time.