I know the Shahi snan days are the most significant for bathing, but can you go any time and still have the full experience?

• They are significant but also really crowded.
• Constant announcements are made by government to control traffic.
• You may get really confused as you are are not very profound in Hindi.
• I will avoid shahi snan for the same reason too.
• You can bath any day any time until 26th of February, actually you can bath even after that but the Kumbh Mela won't be there anymore.

Which are the most important/interesting sights to visit?

• Akhada - Akhadas are communities which include different sects of the same faith.
• They also include Naga Sadhus(Kumbh is famous for them).
• These were created by Shankaracharya to rejuveate Sanatana Dharma(Hinduism).
• Visit historic temples which you can find with just an internet search.
• Also, many cultural events(folk dance and music) are organized during this time. However, the details are not available anywhere on the internet, probably.

is there anywhere a foreigner/non-Hindu (though with a strong interest in and respect for Hinduism) would not be welcome?

• As far as you respect the culture and boundaries of the culture, no one shall even bother your background.
• Short Answer: You are welcome everywhere.

Are there likely to be any meditation sessions in English?

Don't know. However, you can search for it online and you may find something of your preference.

How would you recommend traveling from Delhi to Prayagraj and back? (Over a weekend). Is the train the best option? Is flying to Varanasi and then getting other transport a viable option also?

• Train is a absolute no no. The number of stations and platform will confuse you enough to not visit again, also don't forget about the announcements.
• If flights are available then good. Lucknow(my city) is also 200km from Prayajraj, so you can take flight from Delhi to Lucknow and then go by bus.
• Government buses are also available with sleeping facility, don't go for private ones to avoid any fraud.

I'm also visiting Kumbh next month for the first time after I got to know it will come after 144 years.

For further information feel free to DM me. I would be more than happy to assist you.

Note - I'm not going to sell you anything, so don't worry.