I can only help with one of the questions...but GarryRF, one of the TG Gurus, lives in Liverpool and can certainly help with that part.
For Stonehenge without a car, there are really two main options. There are a number of tour operators who run coach tours from London to the site; or you can take a train from London to Salisbury and take a bus from the station to the Stonehenge visitor center.
The visitor center is new since I was there; it's about 10 minutes by shuttle from the stones themselves, and has a museum, shop and cafeteria. Click HERE for the website of English Heritage which operates the site and the visitor center. There's a lot of useful information on the site, including planning your time at the site.
This LINK is to the operator of the bus that runs from Salisbury's rail station and bus station to the visitor center; the cost is 24 pounds per adult, and includes both the round-trip bus ride and admission to the Stones and Old Sarum which is nearby. Feels steep, no?
London to Salisbury is on South West Trains; about 1:20 and 37.40 pounds round-trip; MegaBus has almost the same schedule time and is 10 pounds each way.
Buying a coach tour from London seems to run around 100 pounds, but includes pickup from hotels, a guide on site, etc. Cheaper than doing it by rail, more expensive than by Megabus. Here's a LINK for one typical operator.
I've alerted Garry to your question, so I'm sure he'll be posting here soon! He and TravelRob also have some Liverpool blogs posted here.