Unfortunately there's no quick fix to getting fit.
When you say your shins are super tight - make sure this isn't shin splints, since it may put you out of the run altogether if you continue to train on it without proper care - you may want to google or youtube some videos to double check and be sure to stretch/rest them appropriately.
In terms of calves, for me at least, the first time I blitz a muscle is always the worst in terms of burning sensation and recovery. If you are able to push your calves to their limit before the race day (while leaving enough recovery time before the big day!), I think it would be beneficial. This means you should be aiming the best you can produce already.
The only thing you may possibly be able to 'rush' is your mental game since it will probably be the biggest factor for you on the 5k run. Always remember that your mind will be telling you that you can't go on or your body is too sore to continue. The best way to overcome this is with a positive training partner (thus the use for personal trainers) or if you can't find anyone suitable in time, try and set goals for yourself. Goals can be either time related or length related, but make sure they're actually challenging.
Remember to tell yourself - pain is temporary. You'll probably have sore calves for a day or even a week after, but after that week you'll forget that the pain even happened. What you won't forget happening is your gained fitness since it will stay with you as long as you maintain it. Also, any time you begin to walk or stop moving when you could have actually run another kilometre is simply you cheating yourself, no one else.
In terms of running every day - you need to listen to your body and see how it feels - but I would advise against training every day. You can stay active daily meaning you can still do other activities that keep your heart rate up but if you run one day, your muscle groups used need time to recover. If you attempt to run while these muscle groups haven't recovered you will start to fall into over training which will actually bring more bad than good.