A boots on the ground question

I guess I can't give input on the exact order, but I assume you will have a car? That is a lot to do in a week relying on public transport.

Public transport. I know, that's why I was hoping maybe someone who had done it, lived in the area or had a better logistical sense than I could give me some input. I can and did go to rome2rio.com but having a hard time envisioning a good strategy.

Bodiam Castle is the bus #349 from Hastings, every 2 hours, Tunbridge Wells train from Hastings, Leeds Castle bus #510 into the grounds Saturday only from Ashford railway station, every 90 minutes. Use the train Hastings to Ashford.
From Hastings, Hever Castle is train to Tunbridge Wells, bus #29 to Uckfield every 30 minutes, (50 minute ride), hourly train to Hever, short 30 minutes or so walk.
Bus 29 starts at Brighton.

Sissinghurst is the hardest - train from Hastings to Staplehurst, changing at Tonbridge or Ashford, then bus #5 hourly then 30 minute walk from Sissinghurst Village, not all of which has a sidewalk.
Or prebooked taxi from Staplehurst station.

Alternatively bus #349 hourly from Hastings (or bus #1066 via Battle) to Hawkhurst then bus #5 (which starts in Hawkhurst, before Staplehurst).

that will be a challenge without a car.

I recommend using the steam train to visit Bodiam Castle, it's the most fun way to travel: https://kesr.org.uk/

You can get a bus to Tenterden and then travel the rest of the way by steam train.

I'd guess that the castles and Sissinghurst don't open until 10 am and, as you say, close at 5pm although there is often a 'Last Entry' time that is earlier. With a car, you could probably visit two of these in a day but without one I think you'd be pushed to do more than one.