Generally if you need to refer to the story, people will count the card as failed. I believe Heisig may say that referring to the story is okay when trying to write the kanji in his Introduction, but this piece of advice is almost universally ignored.

Even if you fail some cards, continue to add new cards. If you get 80%+ right, then you can continue to add new cards without any worry. In fact, you should add new cards every day to keep the habit, even if it's as few as 5 on a day you don't really feel like reviewing.

I'm assuming you're using the "Review" tab on the website for all your cards? Because I'm not quite sure what you mean by "Do I need to add cards even if I failed them (I have't added new cards since I thought that restudying the failed kanji counts towards that?)" All the cards you've seen so far should be reviewed using the "Review" tab on the website (or with more work you could set up Anki which does the same thing but is more flexible).

The stories should be mentally recalled when you see the keyword and you should ideally be able to write the kanji by only mentally thinking of the story.

Again, I may have misunderstood part of what you said, but I hope this helps.

You can add as many cards per day as you like.  I would say the main limitation for most people is the time available for study.

You don't need to memorize the story if you can remember the kanji without it, but most often, I need to remember the story, or at least its outlines.

I'm a bit confused by what you said, as well.  The SRS system should repeat cards "fairly soon" if you fail them.

You study kanji because you want to 1) write Japanese 2) read Japanese, and with this in mind you need to think what it takes to do those things.

Re: writing. Let’s say you know what the word けいざいmeans and now you want to write it in kanji. Most likely your thought process will follow this path: “けいざいmeans economics. The mnemonic for economics, using Heisig keywords, is: mantra, finish. What’s the character for mantra? What’s the character for finish?” As you can see, you can easily get to the shape of the characters if you can produce strokes for mantra and finish. If you remember those without any need for mnemonics it’s great (there won’t be many characters like that though). If in order to recall the strokes you need to go with Heisig primitives first: mantra=thread+(toilet paper=croch+soil), finish=water+(adjust=pattern+two poles), it’s good too – from primitives you get to the strokes. Can’t recall those? You still have a story for mantra and finish – once you recall the story, you will identify primitives in that story and from the primitives you will produce the strokes. So, you have to be able to recall at least a story in order to be able to write a character – or, in other words, if you can’t recall at least the story then you can be sure that you will fail to write the character. Hence fail to write the word.

Re: reading. Some will argue that in order to read Japanese you don’t need Heisig – in a way I don’t think this is  completely wrong perspective. And there are people out there who do have they flash cards set up to have a character on the front and a keyword on the back. Heisig helps them to deal with intimidating character set, and teaches them how to distinguish one character from the other. But they are left with reading ability only so they take only half of what Heisig has to offer. With reading activity you need instant recognition (which is ok because it’s way easier to recognise a character than to write it) and with time you will be able to recognise many of the kanji with resorting to any kind of story – if however there is a character that you have problems with, you start a process that is similar to what happens when you write it but in reverse direction. You start identifying primitives, from primitives you recall the story, from the story Hesig’s label/keyword, from the kewords you deduct meaning of the word.

Ahh, the last think, けいざい is 経済