How many features should I take out of 515 features?
Disclaimer:  I do not know enough about MATLAB to answer the full question but I'll have a go at answering the rest
1 - How many features should you take?
If this is a predictive problem you're trying to solve, how about you try and assess the accuracy of your predictions by using features sets of 7 and 10 respectively? You might find that there is very little difference in predictive power so taking 7 might be OK.
NB: I am quite surprised that your feature selection algo selected such a small proportion of your available features - you're right to want to dig a little deeper on what's happening under the hood..
2 - Will any ML algorithm be able to capture the characteristics of the data with just these 7 features?
If your feature selection algo is working fine then there's all the likelihood that a ML would be able to characterise the underlying structure of your dataset relatively easily.
There's a number of problems out there that have a large amount of explanatory variables but are best characterised by a select few (for instance GDP is usually relatively well explained by what it was last quarter, with all other related variables having small impacts on prediction accuracy).
3 - MATLAB QUESTIONS: TBC
Sorry I can't answer all your questions - maybe someone will come by with MATLAB knowledge and we can roll our answers together. Otherwise you might be better splitting this into two questions to get more detailed answers on the MATLAB algo.