Best ML model for predicting yearly data with many blocks? I would like to forewarn by saying I am by no means an expert in this topic and I apologize if I mix terminology, phrased the question wrong, or any information is incorrect. 

I would like to use a machine learning model for the current dataset that I have gathered. The dataset has about 8000 blocks (or entities), and each of these blocks has 7 features. Each feature has around 10 yearly values, which are counts (ex: 2005=1, 2006=2 ... 2014=0, 2015=3). The long stretch goal would be able to predict the 11th, and even 12th, yearly value for each feature. 

I would really love to apply machine learning to this dataset. I've looked into the following because they seem to be the best solution to my problem: linear regression models, LSTM neural networks, and time series forecasting, although I am still unsure of how my dataset can fit into these certain models.

Please let me know if I can offer any more information!
Your data consists of time series. Recurrent neural networks, RNNs, are applied successfully to problems like this data. Its structure is appropriate for you. Especially, you can use LSTM which is very powerful for time series data. Also there are a lot of papers in arxiv about its capabilities, pros and cons.

I think, linear regression can fail for your task.

I suggest you to read this and this articles to understand how LSTM works and what you can do with it. Then you can decide is it appropriate for your problem or not.