If you are new to running, try it for long enough to see whether you like it. Races should be viewed as an extra. They give you a chance to see whether you have improved, stayed the same or perhaps slipped back a bit, but they are unlikely to be useful as long run (forgive the unintended pun) motivation because you will spend far more time training than you will racing.
You see 5K and 10K races because those are major events in international competitions where the dominant measurement system is metric rather than imperial. One kilometre (1,000 meters) equals approximately 5/8 of a mile, so 5K is 3.1 miles and 10K is 6.2 miles.
Seven weeks is likely enough time for most people to become comfortable running 5K. Just remember good race etiquette. If you're not fast, start near the back of the race pack. Most races these days have chip timing so you will know how long it took you to cover the course, and you don't want to make life difficult for those trying to reach personal goals.
Although this is a website devoted to running, do not rule out the possibility of walking. Walking quickly can be demanding. I have seen racewalkers keeping pace with mid-pack runners. If you try walking quickly - remembering to keep one foot on the ground at all times - you will be surprised at how much effort fast walking demands. Unless you are comfortable running the race distance in training, ask yourself whether you would be better to set a goal of completing your chosen distance first, and worry about racing it later.
Keep in mind that there are some injuries which stop people from running but still let them walk. It really is a valuable alternative. Normally the transition is from running to walking, but nothing says you cannot go the other way as well.
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