- Is merit something that follows my thoughts and actions as a consequence?
Yes, like demerit is: Dhp 1 & 2 when you think and do.
- Or is merit more like an acquired disposition to act and think in certain ways?
Yes, like demerit is: Dhp 1 & 2 when you ponder about thinking or doing
- Or perhaps it's both an acquired disposition and a consequence?
Yes, like demerits is: Dhp 1 & 2 and you meet even the fine nuance that makes the Buddhas teaching different for other kamma ideas. That it is the past intentions and the present intention in the moment, which forms you future experience future.
- Are we born with a certain "amount" of merit due to karma?
You are born because of kamma, yes.
- How does a person know that s/he is accumulating merit?
Only when a person is certain trained, he can reflect it such was at least meritorious. Puñña means always giving, letting go and leads always to release. So when ever there comes up a certain release and relaxation, then there was merit as its cause. Its like having done work. If you have done it correct and right, you lay down relaxed. Of course you can lay down even if not done or not correct, but that possibility is just a making use of previous fruits, credit(s). See as explaining the Lonaphala Sutta: The Salt Crystal
A nice overview is "Ten Ways of Making Merits" and if you like to understand it better, Atma suggest you to get into a little "Abhidhamma in Daily Life", by Ashin Janakanbhivamsa, which is also good to read and with samples.
What you, how ever should not do, is to try to trace kamma up and down, such, as the Buddha told, will make you crazy. On the matter how kamma actually works and is used to escape, a read of Kamma & the Ending of Kamma and Kamma, A Study Guide by Ven. Thanissaro Bhikkhu, might be useful.
And since you are doing here merits "listening to the Dhamma", as for every merits, that you are sure to make not just an exchange or keep anything, don't forget to share your merits, dedicate them to ever living being, which is able to rejoice with your merits you have done. Dedication of Merit
(Note: this answer has not been given with the agreement to be means of trade or the purpose of/for trade and/or keep people trapped and bound. How you handle it lies in your sphere, but does not excuse the deed here either.)