Ruby, 4 tasks in 288280 bytes (non-competitive)
This is just an attempt, I'll keep making tasks later (and hopefully golf the existing ones).
Task 1
a=gets.split.map &:to_i;p a.all?{|e|e<a.inject(:+)-e}
Task 2
p gets.split('')chars.permutation.map{|a|a.join.to_i}.min_by{|x|(x-1e6).abs}
Task 4
n,b='EFHILT',0;gets.split('')chars.map{|c|b+=n[c]==p ? 0:[4,3,3,3,2,2][n.index c]};p b
Task 5
a='Alex A.';gets.to_i.times{|i|i%2<1 ? a.sub!('A.','.A xelA'):a.sub!('.A',a)};$><<a