Cell phone in England do NOT have international calling

No, you do not need to buy a phone, nor do you need to rent one. What you need is a phone which works on the European system (GSM) and which is "unlocked". You may already have one. Then you buy a "SIM card" once you get to the UK and put in the the phone. The SIM card is your connection to the service provider and provides your telephone number (a British number).
You need a "pay as you go" SIM card which you can top up over the internet if needed. Most are.
All SIM cards allow you to dial any number, within the country (UK) or international, and to be called from anywhere in the world. I have not heard of a service provider not allowing international calls, that must be a peculiarity of your country. All European SIM cards work in all other European countries, but check the rates.

AFAIK you can only buy a UK SIM card in the UK.
"download London pass and Oyster Card before I leave home" - you definately cannot "download" an Oyster card. It is a physical card which you use to enter or exit the public transport system in London (buses, tubes, rail). It is a prepaid card which you top up with cash before it gets empty. Details here: https://tfl.gov.uk/fares-and-payments/oyster/what-is-oyster

My phone does NOT accept a Europe sim card so this answer does not help me.

I have a Samsung Galaxy Grand Prime with Straight Talk which does NOT allow international calls. If I am missing something you'll have to tell me as I am no technophile :(

I use a Samsung Galaxy Phone from Straight Talk. Can I still purchase a European SIM card and use my current phone? Will it mean a new number ? I am not technical so need help with this. Don't know how to use Face Time or Skype to make calls.

First, rather than ask a question, you probably need to go off and do some research, whether you are technically minded or not.

However...

Your Galaxy phone takes a micro SIM card and from what I can tell is a GSM phone that will work in Europe providing the phone is unlocked.

StraightTalk is a pay as you go plan with no contract, so I can only assume it is unlocked, it has nothing to do whether your current pay as you go allows international calls. Talk to Straighttalk and explain what you want to do and see if they tell you "won't work" or "no Problem" You would need to remove your Straighttalk SIM card and replce it with a UK SIM card.

You can go to a Carphone Warehouse store once in the UK and they will help you get your phone going, you will get a new SIM card, a new phone number, some data and some minutes/text, can buy more as you need.

Given you describe yourself as not quite technical and you can not get your SIM card before you get there...give up on doing the Oyster card and london pass on your phone, just do it the old fashioned way, get an oyster card once there and order the London pass and either have it sent to you or pick it up there.

To answer your original question, renting cell phones stopped being viable about the same time as dial up modems did. While you can probably still find a rental if you insist, for two weeks it's cheaper to buy (rentals were about $10 a day).

If your phone is locked, you will indeed need to buy one in England. Carphone Warehouse is the easiest place. What kind of phone you need depends on what you want to do. Calls and texts, or Internet as well?

If your phone is unlocked, you can just get a UK SIM card. How to find out if your phone is locked? Find a friend who has a phone on a different carrier, which takes the same size SIM card as yours does. Insert that SIM card into your phone and turn it on. If it almost immediately says "Enter SIM PIN 2" or something similar, and refuses to boot up further, your phone is locked. If it boots up and starts searching for a network, your phone is unlocked.

I don't think anyone has yet answered your question "Can I still purchase a European SIM card .... Will it mean a new number?".
The answer is Yes. It is the SIM card which defines who your service provider is and what your phone number is. The SIM card is like the wall socket you plug your home phone into. If you plug a new phone in, it still has your number, but if you plug your phone into your neighbour's phone socket you have his number and he pays the bill.
A SIM card will have a phone number from the country you bought it in. If you want to call the USA you must dial: +(country code)(area code)(number)

If you need to buy a new phone as well, a very basic phone (not a smart phone) will cost about £20-£30.