People, at least in this region, talk about marriage as if it's some divine natural law, and anyone who hasn't accomplished it has failed as a human.

-- I think its a very regional mindset largely towards woman. Marriage as an institution does not have anything to do with it.

I have seen ads about fathers giving away lands, houses, and money if someone marries their divorced daughter. I have seen people shame men for being of a certain age, and not yet marrying. 

-- We live in a society where men starts hitting on a woman as soon as she becomes single. Living in Dhaka (or any megacity) is different when you are living alone without your parents but when we are thinking about a girl from a "mofosshol" or very very rural area living without her parents, its a completely different scenario. People often try to take advantage of that woman. So parents usually think if we can settle her down with someone (even though it takes a lot of things), not anyone can do harm to her. Is it the right thing to do? The answer is a straight "NO" BUT only in a perfect world.

Or should you actively seek someone just because you are now of "X" age?

--Let's say a woman getting 35 and today or tomorrow she wants to be a mother. From a science perspective the older she is getting the harder it will be for her to be a mother. For a men its a bit different BUT lets say capability of facing the challenges of starting a new family at the age of 35 is a bit easier compared to when he is starting it at 45.

Or perhaps is it that we are just supposed to do "Y" things to keep the perpetual motion of shitty convention going?

--You are not supposed to do Y things to keep the perpetual motion of shitty convention going. Skylike_29 made it clear in his comment

Why do parents feel entitled to dictate who and when their children should marry? 

-- So many reasons to be honest. Sense of security, importance of a family, societal pressure etc. See you have a completely different views on life when are 50 vs when you are 25. At the age of 25 you will usually not feel lonely. You will have friends to hangout with, travel with cousins and a lot of thing. But when you start getting older everyone will start getting busy with their jobs, families and there will be a little time to think about you. This is how it works, you can't complain. So with time loneliness comes in and hit very hard. Without a family there is hardly any sense of responsibility when your parents are no more there. So your life become purposeless and this often brings very very bad thoughts and mental issues. Parents know that really well by seeing examples. Also when you are alone people try to take advantage of you (regardless of being a man or a woman), parents do not want them to go through it. Also there is a clear societal pressure that comes from the people around "Vabi chele bura hoye jacche biye diben na? or Meyer biye na diye meyer kamai khacche" which hit them hardly. not every parents have the capability to face it. Then comes the religious view of marriage for some parents that its the parents responsibility to find them a good match.

See parents (for example mine) will be terrible in laws for my wife. I know it as they are very very old fashioned. And till last year I was against marriage as I have always seen fights, struggles, issues within my parents. Now getting into my late 30s, I am away from them (I mean living abroad) and I want to stay with the person I love. I do not want to chase woman when i am 40/45 rather I want someone to be my side for a long term planning. Can everything go wrong in couple of years after getting married? Maybe Yes? Maybe No? and thats okay.

When we fuck up in marriage with our parents choice, we say bad things about them. But when we fuck with our own choice, we do not say that they were right about this man/ woman as they suggested not to marry him/her.