I'm going to rely upon the SWEBOK (Software Engineering Book Of Knowledge) to guide my answer.
In the Software Requirements* chapter, we're provided with more formal definitions of the terms needed for your question.
Functional requirements describe the functions that the software is to execute; for example, formatting some text or modulating a signal. They are sometimes known as capabilities.
Nonfunctional requirements are the ones that act to constrain the solution. Nonfunctional requirements are sometimes known as constraints or quality requirements.
I point those out because there appears to be some overlap in the way you have been requested to generate your document. You stated: A SRS should only have functional, non functional and constraints and based upon the SWEBOK / IEEE definition you can see that non-functional requirements and constraints are one and the same.
And while some may protest of overkill, there are three main documents related to requirements work.
System Definition Document This document (sometimes known as the user requirements document or concept of operations) records the system requirements. It defines the high-level system requirements from the domain perspective.
System Requirements Specification In this view, system requirements are specified, the software requirements are derived from the system requirements, and then the requirements for the software components are specified.
Software Requirements Specification establishes the basis for agreement between customers and contractors or suppliers (in market-driven projects, these roles may be played by the marketing and development divisions) on what the software product is to do as well as what it is not expected to do.
So the problem with all of that is it appears clear cut when explained, but it ends up being muddy when implemented. To help focus my answer, I'm going to ignore the System Reqs Spec as it's a huge topic and not quite necessary for your question.
The primary challenge you're encountering is that you are mixing your SDD and your SoftRS (software) documents. Since the two documents are communication bridges between three camps, it's natural and unavoidable that you'll see some overlap between the two. The SDD is a bridge between the client and the analysts, whereas the SRS is a bridge between the analysts and the developers.
So how do you draw the line between the SDD and the SRS? Let's go back to the System's Engineering trinity:
And they really should go in that order because when we look at the V-model, we use those three in order to work our way through the layers. You start with the what & why at the system layer. As you capture those, you start generating ideas for the how, which goes into the next layer below in the V-Model. "Eventually" you finish up with a particular layer and you start revising the layer below in the same manner. The difference on the lower layers is that the upper layer's how becomes this layer's what.
Blah, blah, wall of text, let's tackle your questions.
FR 1.1 : Have I incorporated a design idea here, would " the system shall have a registration form" be a better functional requirement?
I'm going to start off with "meh" for an answer. "shall display" vs "shall have" on a form that requires user interaction is kind of pointless. If an argument erupted on a real project over the semantics between those two forms, I'd consider that a huge red flag and try to bail on the project. Seriously.
"Shall display" implies "have" whereas "shall have" implies "display" in this case. To be fully pedantic, you would need at least "shall have and display" to make it clear. Any of those three variants is fine though.
F1.2 ,2.3 : Is this not singular? Would the conditions be better written for each its own separate requirement
They are certainly related, and it gets tricky on how you want to express those. I rely upon team norms in this case as I have seen it go either way with equivalent results. The important part is to be consistent in how you handle compound requirements like this.
FR 1.4: Is this a design idea? Is this a correct functional requirement or have I incorporated non functional(performance) in it? Would it be better if I written it like this:
FR1 The system shall display an error message when check is unsuccessful.
NFR: The system will respond to unsuccesful registration form checks within 1 seconds. Same question with FR 2.8 and 2.9.
I would probably break these out, yes. Here's why: The first is providing function (display error message) while the second provides the constraints (specific amount of time) in providing that function. Constraints often change, but the functionality doesn't. By moving the constraint to its own requirement, we can get up front agreement on the functionality and then deal with the constraints (how fast was the message delivered) when we get into implementation.
FR2.3: The system shall check for "completeness", is completeness here used ambigiously? Should I rephrase it?
Your close to explaining completeness with the list of items you provide. However, you could be more clear in specifying what determines completeness or not in each of the items you listed.
FR1.2: I added that the system shall require a "Captcha code" is this a functional requirement or does it belong to the "security aspect" of a non functional requirement.
Depends upon what level of your design you are talking about. At a high level, it's a non-functional requirement because it's a constraint. At a low level, it's a functional requirement because it's specifying a function to be provided.
* SWEBOK is currently in a review state for the next release. Except for the main link, some of the additional links may go stale.
I'll speak to your examples.
The first example of a "user requirement" is more like a wish or "feature." The way you can tell the difference between a feature and a requirement is that there's enough detail in the requirement to make it testable. Requirement 1 is not testable because, well, it's a wish. "I wish that the system had some reports for the managers." How do you know that the requirement has been achieved, that you can declare success?
Requirement 1.1 is testable because you can wait until the last working day of the month, and see if a report is generated on that day (or you can inject dates into the system and observe its behavior).
Requirement 1.2 is testable for the same reasons.
Neither system requirement, however, tells you what the reports should look like, how the data is laid out, or how the calculations are made; they only describe the reports in general terms. In practice, there will be a Software Design Specification of some sort that tells you in detail what these reports will look like.
Best Answer
There are many different ways of categorizing requirements. Without seeing the exact text that you are referring to (and Sommerville has written several editions of Software Engineering, with the most recent being the 10th edition in 2015, and then Engineering Software Products in 2019 as a replacement), it's not possible to explain exactly what the author means. However, I can speak about requirements engineering in general.
User requirements, system requirements, functional requirements, and non-functional requirements aren't equal levels of abstraction. There are different ways to categorize a given requirement.
Functional versus non-functional (sometimes called the "quality attributes" or "quality requirements" of a system) is one way. Functional requirements describe behavior, usually in the sense of inputs and outputs. Non-functional requirements describe the overall operation or properties of the system under design.
The distinction between user requirements and system requirements is more about the source of the requirements. User requirements specify what the users and customers expect from a system, in the context of the user's domain. System requirements exist in the solution domain and describe what the software must do.
Often, system requirements are derived from and traced to user requirements. Both levels of requirements - system and user - may have both functional and non-functional requirements. In that sense, yes, it's safe to say that system requirements are an abstraction over functional and non-functional requirements or that user requirements are an abstraction over functional and non-functional requirements.