UK Visitor visa to visit my girlfriend refused under V4.2 (a) (c) & (e)

Lets deal with question set 1 first. All of the statements in the refusal are aiming to tell you "You are welcome to apply again, but unless you fix the problems the next application will be rejected too." You can’t just repeat the application and hope for a different result. It’s standard statements that are always made. Some people think it’s like a lottery, and if they just submit the same application again they might win this time.

For the second question set, the calculation error is not a big issue. It’s not worth trying to get the decision overturned because of it. You can point it out in your next application, in notes.

Now for the questions you didn’t ask. There are two serious issues with your application:

1. The officer can’t find evidence to back up your statements about income
2. There are unexplained deposits in your accounts.

(The unexplained deposits may be from your income but the officer can’t verify that).

You need to find better evidence that the deposits in your accounts actually do come from your work. For example, if your invoices show Customer X paid you a certain amount for some services, then there should be a deposit to your account for that amount from Customer X. Also any substantial deposits need to have an explanation. Any amounts you received that are unexplained will be treated with suspicion. People sometimes try to "pad" their bank balances by borrowing money, and any unexplained deposit may look like that to the officer.

Your comments about "the profile" are reasonable. You do indeed look very like the kind of person who would fail to leave the UK after your visit. You have, on paper, a huge incentive to do that – a girlfriend in the UK, a job you can probably do remotely, no other ties to your home country. (Joint ownership of property is of little value – you could continue to own it and receive income from the UK.)

But the really important question you haven’t answered for them is "How can you afford to take four months off work?" Remember that you are not allowed to work, even remotely, while in the UK. Taking that time off, on top of the cost of flying to the UK, would wipe out almost all of your savings. Most people can’t afford to have no income for four months. And if you can live without income for four months (for example because your girlfriend is earning enough money for both of you), then maybe you can live without income for twelve months, in which case you have another incentive to overstay your visa.

Question 1

• (1) This really boils down to “you’ll be refused again unless you address the refusal reasons credibly.” UKVI will consider additional evidence, but IMHO a 6 page explainer letter is far too long. Put the key points you want to cover in the ‘additional comments’ section at the end of the application and anything else essential and relevant in the cover letter. Imagine you are the ECO making the decision: would you carefully wade through 6 pages of minutiae and long-winded explanations as well as the application & supporting documents?

• (2) If UKVI made a material error, you can draw their attention to it in your covering letter. Consider though, whether the deposit balance information you provided was crystal clear. If it isn’t easy to understand at a glance, find a way to improve it eg get statements that clearly show the totals, or can you highlight the relevant figures to tie up with those stated in your application?

• (Q3) Convincing them that you’ll leave given your profile might be hard. What ties to home did you demonstrate in your previous, successful, applications? Do you have any work commitments after the planned visit or dependents in your home country that would require your physical presence there? Have you considered having your girlfriend visit you this time rather than you visiting her?
  It seems UKVI weren’t satisfied as to the source of your bank deposits. Does all your financial information reconcile? For example, payment amount X from invoice A shows up as X in your bank account on a credible date and with the payer’s reference?

UKVI picked quite a lot of holes in your application. IMHO it would be worth giving serious consideration to getting a ‘doc check’ before re-applying, as recommended in the canonical answer on this type of refusal UK visa refusal on V 4.2 a + c (and sometimes 'e')

Edit: a note on remote working
The Visit Guidance published by the Home Office says:

“Visitors are permitted to undertake activities relating to their employment overseas remotely whilst they are in the UK, such as responding to emails or answering phone calls. However, you should check that the applicant’s main purpose of coming to the UK is to undertake a permitted activity, rather than specifically to work remotely from the UK. Where the applicant indicates that they intend to spend a large proportion of their time in the UK and will be doing some remote working, you should ensure that they are genuinely employed overseas and are not seeking to work in the UK.”

Source: https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/1019544/Visit.pdf

Convincing UKVI that you are ‘genuinely employed overseas’ looks hard to do for a self-employed individual who can work remotely for months at a time.