1. It usually does make sense to check the scale reliability in your sample as the reliability coefficient by definition is a variance-dependent statistic (reliability = true score variance / observed variance). Therefore, the reliability coefficients reported for a given scale in different populations may may vary, and estimated from the previous literature may not apply to your population/sample.

2. If you deal with a multidimensional scale with multiple subscales/subdomains, it does not make sense to apply, for example, Cronbach's alpha to the overall scale score (combining multiple subscales). The reason is that Cronbach's alpha is based on the model of (essential) tau-equivalence of classical test theory (CTT). Tau-equivalence implies a single-factor (unidimensional) model with equal factor loadings. Applying alpha to a multidimensional sum score is meaningless and can also be highly misleading.

3. It is not surprising that coefficient alpha would be higher for the whole scale since alpha is partly a function of the number of items (one of the fundamental laws of CTT is that all other things being equal, longer scales are more reliable because errors of measurement have more opportunity to average out). When you look at subscales, these obviously consist of fewer items and are therefore almost guaranteed to produce smaller alpha values. However, as I stated in (2), alpha would be more or less incorrect/misleading/meaningless if applied to an overall (multidimensional) scale score. Low reliability of the subscales may be due to having too few items (or unreliable items). I would only delete items if they violate the implicit unidimensionality assumption. This can be best assessed by applying confirmatory factor analysis (or IRT models) to each subscale, that is, by testing a single-factor (unidimensional) model for each subscale.