Though you have divided your question up into 6 parts, I am combining the information in the answer, as they all link together. Also, as can be seen in the reference list (below), the resources come from the same primary author, but from different aspects of his studies of this phenomena (these are all paywalled, so university access would be required - but I'll paraphrase the main points).
Firstly, a bit of terminology, a Nor'wester are storm gusts between 11 ms$^{-1}$ and 42 ms$^{-1}$, above which are referred to locally as being 'tornadoes' (2). These are collectively known as Severe Low Convective Storms (SLCS) by Yamana et al. (1),(2),(3). The main areas affected are across Bangladesh and north-east India (3).
SLCS generally seasonal, usually occurring during the pre-monsoon months of March to May, peaking in April, usually in the evenings (as you have observed). Although rare, SLCS can actually occur at any time of the day and at any time of the year (2). Records over several years do not show any particular obvious trend in occurrence and frequency of events that occur each year - though they average at around 145 occurring each year (2).
Yamane et al. 2010 (2) note that the most frequent occurrence is in the Dhaka district (north-central Bangladesh) and trends in a northerly direction during the pre-monsoon. However, the authors caution this observation as being the result of the amount of people recording these occurrences in other districts.
Yamana and Hayashi (2006) (3) explain that the SLCS are a result of regional atmospheric thermal instability, where the convective available potential energy (CAPE) is high (greater than 2000 Jkg$^{-1}$) and the shear is moderate (greater than 10 ms$^{-1}$, and that the phenomena also coincides with severe local storms across the Indian Subcontinent.
Yamana et al. (2012) (1) elaborates further on what causes the atmospheric thermal instability, with a synoptic analysis of a SLCS, describing the phenomena as a confluence of (note, the authors use AGL = 'above ground level' for the following information):
- Moisture inflow (increase in humidity) from the Bay of Bengal, enhanced by a south-west wind intensification at 950 hPa (approx. 500 m AGL). This increases the amount of water vapour available.
- A temperature increase, due to westerly winds, across the region at 800 hPa (approx. 2 km AGL), leading to
- A trough leading to an enhancement of cold advection at 550 hPa (approx. 5 km AGL) from the north west.
This, in part, gives a reason for its name and confirms your 6th point, that it blows in from the north-west.
References
(1) Yamana Y. et al. 2012, Synoptic situations of severe local convective storms during the pre-monsoon season in Bangladesh, International Journal of Climatology
(2) Yamana Y. et al. 2010, Severe local convective storms in Bangladesh: Part I. Climatology, Atmospheric Research
(3) Yamana Y. and Hayashi, T. 2006, Evaluation of environmental conditions for the formation of severe local storms across the Indian subcontinent, Geophysical Research Letters
