Is a Railcard worth it?

A railcard will cost £30 and give you 1/3rd off rail fares. With those 3 trips I guess it would be marginal but you need to do the sums to see if it's worth it.Buying Advance tickets for the second two trips might be cheaper as long as you are happy to travel on a specific service.
A Senior railcard is for anyone 60 and over. If there are two of you (of any age) then a Two Together railcard will be cheaper as you only need one whereas you would need two senior cards.

Any particular reason for going to Paddington?

A railcard costs £30 for a year and gives 33% off so you need to be spending over £90 for it to be cost effective. You need to work out the cost of your intended trips using dummy bookings. www.nationalrail.co.uk

As suggested above, many UK rail fares vary wildly, depending on how early you buy the tickets. So when you figure out your potential ticket costs for the purpose of calculating how much you might save with a rail card, you need to check fares for future dates that approximately match when you would be buying tickets for your trip. In other words, if you want to retain a lot of flexibility and buy your tickets very late (but day-of-travel tickets can be horribly expensive), you would look at prices for tomorrow or Monday. If you are locked into a specific date and time for the trip to Dover, you should check prices much farther out.

If you book a reasonable amount of time ahead you can get Heathrow Express for £10- then get railcard discount off even that early bird fare, so that is about £3.50 to start with (as long as you travel after 9.30am on Monday to Friday) towards the savings you would make.
The Cheapest way to Hastings is usually from Victoria and takes only 15 minutes longer. Advance (train specific) fares start at £7 (railcards further discount that by 34%) or for a fully flexible fare on that route after 10am buy a Southern Rail Day Save ticket for £25 (no railcard discount)-https://www.southernrailway.com/discounts-and-offers/daysave

if you decide that a Railcard makes sense after you look at the fares and your preference for money for convenience vs. savings for inflexibility, one further decision is based on how many in the party will be travelling together - which Railcard to get.

If there are two of you, a Two Together Railcard is only £30 for both of you. You do need to travel after the morning peak 9:30 M-F, and you need to both be travelling together at all times.

Otherwise there is the Senior Railcard for over 60s, the 16-30 Railcard and the Disabled Railcard (upon proof, and £20).

All of these are valid for 12 months.

I'd agree that if you can predict date and time for those few journeys, Advance (the name of the ticket plus when you need to get it, linked to specific trains and effectively non changeable and nonrefundable) tickets may well come in less than you need to spend to break even.

Can non UK citizens get Senior Railcards? My reading of the National Rail site seemed to imply that you needed a UK driver's licence or passport

Anyone over 60 can get a Senior Railcard. Visitors just need to work out whether paying £30 for one is worth it for your trip.

The reference to driving licence or passport will just be examples of how to prove your age. Assuming you are coming to the UK you will have a passport anyway.