1. How do you keep abreast of rival civ's tech research progress?
Once you have Alphabet and can trade techs the game will inform you of what techs a civ has for trade, what techs you have that a civ might want to trade for, and what techs a civ can research. That last one in particular allows you to easily deduce where your rivals are in terms of techs, at least roughly. Do keep in mind that AIs can trade techs among themselves, too, so never assume that an AI can't be close to something without
thoroughly checking all their options.
2. What do you do when the rival civ's culture influence shoots up exponentially?
Nothing. The only time an AI's culture is a problem is if they're going for a Cultural victory (which requires them to have 3 cities with 50.000 or more culture), or if they're your neighbour and are putting pressure on your border cities. The former case is only a problem if that puts them on schedule to win before you do, the latter is only an issue if you can't take the offending cities from the AI and make them your own. And anyone focussing on culture isn't focussing on key military techs to actually keep their cities safe from invasion, so...

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3. What should be the research path now that I have entered Renaissance era? As you can see, I just went ahead with tech that will lead me to rifle and cannon.
Traditionally people will beeline towards Liberalism, since the first person to research that tech gets a free technology and obviously that's as great a thing to get as it is to deny to any of the other AIs. But the path towards Lib is really only useful if you're going to continue teching, since Lib requires two techs and neither has any direct military value on their own - and neither does Lib itself, for that matter, beyond getting you a free tech. Philosophy is great for Pacifism, which militarily is about as useful as you'd expect (for
getting to the military techs, especially through Lib, Pacifism is nothing short of fantastic. But once you have an actual army on the board, less so). Education enables Universities and Oxford, both of which offer a serious boost to your beaker output -
especially if you've prepared a strong Bureau capital with lots of mature cottages - but when you're dead set on conquering the world with your military prowess before anyone can tech/build up enough to stop you than both of those are just useless. So going towards Lib can end up being a very expensive detour that you could have instead invested in teching military techs the normal way, at times.
There's ways to make going towards Lib more rewarding and less painful, limit the opportunity cost and maximize the benefit, but in your situation it's probably better to just get cannons and rifles and roll over the entire world with them. Rifles and Cannons against Noble AIs researching Guilds and Engineering is going to be a
very one-sided contest, especially when you've got 15 cities.
4. Can you take a look at my cities and see if I'm managing that correctly? I'm not averse to building markets and courthouses to try and alleviate the economy situation - poor move?
Whipping out a Courthouse or two when you capture an absolutely massive city that would just starve down otherwise isn't a completely unreasonable idea, especially when you don't need/can't support any more units so whipping a few more units out of your captures isn't a viable alternative. That said you're usually better off building Wealth than building Courthouses, especially when you're starting to get close to Communism, which will solve all distance maintenance-related problems forever. Or at least until the UN inevitably forces everyone into Environmentalism, but by that point your economy should be able to manage.
4. For all intents and purposes, it seems that I am now left with only one major rival: the Aztechs. I'm keen to hear your thoughts on how to rush headlong to a winning finish.
From the looks of it the best unit that Monty is about to get is Knights, and you have Rifles. If you show up on his doorstep with an army of Rifles (and cannons, preferably) that's not a case of you catching him with his pants down, that's you catching him before the tailor who will make his pants has even been born.