IRCTC website now allows using international mobile numbers but there is a fee to use a foreign number, Rs100 +GST which I believe would work out to be Rs118 which is $2. IRCTC also accepts foreign credit/debit cards now.
There is no point getting a sim card delivered by paying extortionate fees for such services. You can buy an Airtel SIM card at New Delhi airport when you get out of customs there's a counter there. Mumbai airport has a similar arrangement by Airtel.
Airtel has a plan of Rs450 for 28 days and you get 1GB data per day with unlimited incoming and outgoing calls with no roaming charges. Their network is pretty strong across the country so no worries on that front.
If you find it difficult to deal with IRCTC directly you can book your tickets through cleartrip or Makemytrip which are privately run companies with a responsive customer support department who would reply to your email queries much faster as compared to IRCTC.
In case you find yourself without any bookings when you arrive in India, most big railway stations like Newdelhi, Mumbai, Kolkata have a dedicated office for foreign tourists and they have a reserved quota of seats for foreigners. I wont say you are guaranteed a seat but you have a lot better chance of getting seats that you want as compared to locals.
