How to compare Timeseries Sequences?
The answer to your questions depend a lot on the nature of the data represented in the time series. You should ask yourself some questions to better understand what might or might not work, e.g.:


Are the time sequences perfectly aligned?
Are two slightly shifted time series considered similar or not?
Are two time series with the same shape but different scale considered similar?


Normally, the answers to those questions are that series are not perfectly aligned and that variations in scale are also fine as long as the shape is similar. For these scenarios, the classical measure is Dynamic Time Warping (DTW). There are lower bounds for DTW that are very efficient computationally. The research of Professor Keogh might be interesting if you need theoretical foundation for it.

Also, normally euclidean distance and Manhattan distance are not very appropriate for time series due to their sensitivity to signal transformations (e.g. shifts), but actually they are often used in practice.