Is an HOA required to send you a statement?

If they adjust the fees, they have to let you know. Otherwise, no, they don't have to send an invoice every month.

Why would they process a statement for every homeowner in the community and send them to every homeowner? Do you have realization of the cost of all of that, at the expense of the homeowners? Obviously not, or you would not be asking such a question. The HOA dues are set each year and EVERYONE is made well aware of what they are - once. If you are too lazy to read your mail or show up at a meeting, stop into the office and ask. Don't be suggesting an unnecessary and cumbersome and expensive process that no homeowner in his right mind would want.

Probably easiest to just pay the late fee. If you want you could pay it and then sue them for the late fee saying you were never told about it. I kinda think they'll just drop the late fee but if they go to court and say they sent you a bill, don't know why you didn't get it, it'll be difficult to prove them wrong.

They do need to either inform people of what the fees are or make sure you can easily access that information yourself. The problem is, these HOAs are often run by average joe's off the street. They get power and they do what they want.

Not unless the bylaws require them to do so. Go to the next monthly meeting and play catch up. They will likely give you access to the minutes of the past meetings and maybe consider having your vacation home mail forwarded to your permanent address. Or better yet, run to be on the board and get the bylaws changed to demand the HOA to send statements. It's your option as a an owner.

No, they are not required to send you a monthly bill or receipt for your regular dues just like landlords don't have to send bills for the rent.

There is no law that requires them to send you a monthly statement. You know you owe HOA dues, you should know the amount and the due date, so you should pay them without being reminded.

If they changed the amount they should notify you as a matter of common courtesy. If you payed short one month because you didn't know, then you owe the difference. If they are charging you a late fee but they never notified you of the increase then you might have an argument to fight the fee, but not the increased amount.