That's right, H has to be non-empty (easy to check: 0 has to be an element) and you can combine the other two properties into one: it has to be closed under lineair combinations, i.e. if v and w (vectors) are solutions, then kv+lw (k,l scalars) has to be a solution too.
1) 0 is a solution, trivial. If (a,b,c,d) and (e,f,g,h) are solutions, is k(a,b,c,d)+l(e,f,h,h) a solution too? Check it
2) Try it.