As said that is a bit of work and may be totally unnecessary.
The sniff test is not a good name or description for non-UK and what actually happens. You can get small test kits off eBay (and other places that test for combustion gases in the coolant, either litmus test strips or liquid test kits, just as one example. - https://www.ebay.co.uk/itm/123735248113 You can also read up on symptoms but these are not always that obvious particularly on an engine that might not be fully set up correctly - but your engine might be set up reasonably or well, we do not know. Best is to get your dad to explain and show you how to do cylinder compression tests on the engine.
With an old (or any car really) car you should always have all electrics (other than safety ones) switched off if the engine is not running and not turn them on until the engine is running. A good routine to get into is to turn all electric (except safety) items off before you turn the engine off and certainly before you get out of the car. When you get into the car and seated, turned the ignition on, but do not start the engine, check all the warning lights are on that should be and those that go out do go out, whilst you are doing this you can put your seat belt on. Once you have your seat belt on check the handbrake is on and car out of gear and start the engine. When you have started the engine do not turn any electrics item on yet (especially radio and phone) but instead listen, look. smell, feel, the engine and car for anything unusual, if all OK drive off.
I am not sure about your present wiring, if there was a three pin fan switch fitted before and you have replaced it with another three pin switch assuming the correct part was fitted, when it was not, that is one issue that might explain the joined wiring but if it was a two pin before it is more of a mystery to me. The radio and phone added items are another area I would check.
You may not need to get a new battery. If you check the existing battery has electrolyte (acid) "water" covering fully all six cell plates and the plates do not look too damaged or buckled that battery, top up if required before recharging, older cars give more chance for the battery to be recharged to give more useful life (subject to the wiring being correct and not draining the battery unnecessarily. You best to use a suitable charger (maintainer) with low not high amperage or setting, 2-amps, 3-amps, 4-amps, certainly not more than 6-amps for your 62 Ah battery. The lower and slower the recharge then, for want of better words and description, the deeper and longer lasting the recharge, this could take many hours, probably overnight perhaps longer depending on how low the battery is and charger amperage. Car batteries are one of the most oversold car parts, often prematurely replaced as "distress sales" as with you now, preventative charging before they get too low will give much better and longer life. Neglect of battery is the number one reason for breakdown call outs in the UK.
Some notes if you want them, some of the information varies and expands the between the links, some obviously is repeated.
Best check your car battery now - (why?) [ summer ] - https://www.briskoda.net/forums/topic/506654-best-check-your-car-battery-now-why/#comment-5688832
Car battery, now is the time to check it [ winter ] - https://www.briskoda.net/forums/topic/499006-car-battery-now-is-the-time-to-check-it/#comment-5598720
flat batteries on the first working day of each year - https://www.briskoda.net/forums/topic/519903-flat-batteries-on-the-first-working-day-of-each-year
HTH.