I'm not sure whether you're talking about Discrete Wavelet Transform (DWT) or Continuous Wavelet Transform (CWT). Both can be used on discrete signals similarly to DFT and DTFT, I'm not sure if anyone calls it the Discrete Time Wavelet Transform instead. In any case, the dyadic wavelet transform is the non-redundant one (number of samples in $=$ number of coefficients out), and that's what we usually assume by talking about DWT.
It looks like you need to do a bit more reading on the wavelet transform in general, because of your first question. Wavelet transform doesn't work like the Fourier Transform; rather, it's akin to the Short Time Fourier Transform, which is a function of two variables: frequency and time. Similarly, the wavelet transform is a function of scale and time. This means that you don't pick a scale and stick to it. You go through many scales (in dyadic case they increment in powers of two), and calculate the transform for each one.
To address your second question, edge effects are usually not that easy to deal with and there's a plethora of papers on the topic. If you simply want to know what portion of the transformed signal will by affected by them, and it looks like that's what you're asking for, this paper has a good discussion. One thing to keep in mind is that the DWT, like the Discrete Fourier Transform, has circular wrapping symmetry, so dealing with edges is undesirable if you're hoping to get a perfect inverse transform as well. You can look at this paper for a discussion of that, and also a way to eliminate edge artifacts.