Have I been conned by letting agent?

This does not sound good. You may have grounds to cancel the agency agreement. But it depends on what the agency agreement you have (presumably) signed actually says!
We have a guide for landlords here https://landlordlaw.co.uk/openaccess_trails/dealing-with-problem-letting-agents-a-guide-for-landlords/ which is available to members of Landlord Law. The guide takes you through all the things you need to check when you have a dispute with your agents, and advises on your legal remedies.
However, landlords have very limited rights to set aside agreements that have been made on their behalf with tenants - as you can't expect the tenants to know that your agent was exceeding their authority. This is all part of 'agency law' (a specific area of contract law), which is also explained in some detail on Landlord Law, for members. https://landlordlaw.co.uk/the-landlord-law-legal-basics-courses/.

Complaining to their Property Redress Scheme is a good idea, but they will expect you to have tried to resolve things with the agents direct first.

Is the agent registered with UKALA or something similar? If so, maybe they can assist you.

I would do whatever I needed to to to get out from under that contract with the agent. Regardless of what the contract says in relation to termination clauses, I would view the lack of communication and proper information regarding the tenancy and works as grounds enough for me to terminate our arrangement. I would visit the tenants and explain the situation (with proof of ID and ownership). I would ask them to change the rent account details to mine and change the locks. Others may say this is risky behaviour which may expose me to legal action but that is what I would do. I fact I had an agent that was a saint in comparison to your agent; however I found out he had let a couple move into one of my properties prior to signing a tenancy agreement and that was enough for me to terminate. Good luck.

With regards to the clueless gas engineer which stated that a new pump was required for the boiler, but later the company re-attended and found a break in the circuit. You refused to pay for a pump which had been a wrong assessment and the two invoices were nearly half the price of a new boiler.

As you showed him that there was no credit on the pay as you go meters the engineer could not legally assess and re-commission the boiler. Despite no gas supply the engineer went ahead and allegedly assessed the boiler and provided a gas certificate. The engineer is clearly in breach of gas safety rules.

By law engineers have to be registered on the 'Gas Safe Register'. Under the Gas Safety (Installation and Use) Regulations 1998, a gas engineering business must be registered on the Gas Safe Register to legally undertake gas work within the regulation’s scope. If there are any gas safety concerns related to work performed by a registered engineer or information about individuals or businesses involved in illegal gas work, it is recommended to file a formal complaint and report the issues to the 'Gas Safe Register’. (You can do this on-line). The ‘Gas Safe Register’ will investigate this matter, but can’t get your money back. You would need to seek redress through civil action. In court you will need to show ‘Burden of Proof’.

However, no ‘Gas Safe’ heating engineer or business wants formal complaints made about their heating work. The 'Gas Safe Register’ is set-up by the Health & Safety Executive, the legal enforcer of gas safety legislation in Great Britain. You can easily prove the gas engineer could not have followed the Gas Safety (Installation and Use) Regulations 1998 procedure because of no gas supply.

Email your energy supplier and explain the situation. As for proof about the gas fuel credit available on the day the engineer conducted the assessment. The suppliers will be able to obtain this information through ‘key transaction data’, ‘energy usage data’ etc. Once you have the proof it was impossible to comply with the Gas Safety (Installation and Use) Regulations 1998, due to no gas supply, present your findings to the company who did the work. Inform them if they do not refund both invoices immediately, you will report them to the 'Gas Safe Register’.

You might find they will refund your money with no questions asked.

You ask the question "Have I been conned?"; the strong likelihood is yes. However, the agreement you seems to have entered into seems so unusual that I really wonder what you have got into. I've never heard of any agent splitting rental to cover contractors' costs.

Are you sure you've entered a management agreement? Normally the agent would charge all costs (at invoice rate plus any agreed mark up) back to the landlord. Most agents keep a float so that they may instruct contractors.

With respect it sounds like there's a whole story here as regards the agreement you have entered into. Despite this you have painted a picture of total incompetence by the agent and you really need to address this. If I were you I'd get the tenant to pay you directly whilst you sort the agency out - that way they won't get any more of your money.

Legally the agent must keep clients' money separate from office money so they should have clear records and should provide you with proper statements.

Good luck!