There are three types of test......
1. Y-DNA - the Y-chromosome is only passed in the paternal line, so it won't find a match in your mother's ancestry.
2. Mitochondrial DNA (mtDNA) - is passed via the female line and used to trace maternal lineage.
3. Autosomal DNA - as well as a pair of Y or X chromosomes - each person also carries 22 pairs of autosomes. These are gender-independent, and it is claimed that short-duration DNA tracing (within 5 or 6 generations) can be done from both male and female lines. FTDNA calls the autosomal test the 'Family Finder' test.
It's almost impossible to get a DNA test without divulging your name - how are you going to pay for it and have it delivered ? And I'm not sure why you would want to ? The whole point of a DNA test is to find people with the same or similar DNA - how are you going to do this without having a database with people's names on it ? With FTDNA, you can opt out of having your name on the database if you wish.
The accuracy of the results is not a problem. You can already pay for 67-marker DNA tests. What's lacking and is hoped to be improved upon is the number of people in the database. It's a bit like fingerprints - you could employ someone who is a master at finding fingerprint matches, but if you only have 10,000 fingerprint samples, then you're still very unlikely to get any matches with a random sample taken in a pub somewhere. With DNA, it's slightly less complicated, because they are looking for DNA sequences, not a match in the whole sample.
DNA samples, being biological, will decay with time, but your DNA profile is stored numerically, so it can't decay - most labs will destroy the samples, but you should clarify this with them if it bothers you. You might need to take another test if (say) you originally paid for a 37-marker test, and you later wanted a 67-marker test. You have to bear in mind that the DNA profiling that these labs do is not as sensitive or as detailed as the DNA testing done in a crime lab, so it's quite important to make sure you harvest a good number of cells when you do a swab of the inside of your cheeks, because the bigger the sample, the quicker it is to process.
With FTDNA, in addition to sending in your samples, it's also a good idea to send in your ancestry data, and FTDNA can accept this in the form of a GEDCOM file. This is quite important, because if you are matched with another person on the database, you will of course want to know who your common ancestor was. When you first register, you can enter just your 'earliest ancestor' on both your paternal and maternal side, and add to it when you want to. On the FTDNA site, you have your own account pages, and on one page there is a map, where you can show people with the same DNA profiles, and people with the same surnames (if you have joined a name-study group).