Yes absolutely....Mumbai is a safe city but be sure you travel in second class it is extremely crowded but safe coz most of the time women first class compartments are with fewer women so it is convenient but you can travel in the morning times
Mumbai is safe for women, for local trains avoid rush hour, for people from outside mumbai its very difficult. I don't want to discourage you but there had been many accidents. So, go for 1st class, make pass for 1st class, it has less crowd.
Rush hour are morning 6 to 11 then evening 5 to 9 pm.
All the best.
Yes of course it is, just be careful about the timings to make it hassle free by not travelling in the office hours.
Well Dadar is one of the most crowded station mumbai has, so be ready for some action if boarding in rush hours.