Can LSTM Predict The Next Few Days Of Stock Price?
The reason why you find the same dataset everywhere is that financial data is extremely secretive and expensive and nobody is willing to share their financial information because in doing so they will be effectively losing an edge in the market. For that reason public financial data is mostly gathered by academics and not too often.

Second, LSTM can make predictions for the next n-hours, n-days, n-months and even n-years like the article STOCK PRICE PREDICTION USING LSTM,RNN
AND CNN-SLIDING WINDOW MODEL by Sreelekshmy Selvin et. al. and many others show. You just need to build a good training vector and the target can be anything you want. As it's natural in financial markets, accuracy decreases as time increases.