Your first question, "if it is ok to privately email in ex-girlfriend and then lie about it"- No, It is not OK. He should not be emailing "privately" to anyone unless it's someone you mutually know and are aware of. To then lie about it, implies that he is guilty for going behind your back, and then guilty for writing something he knows you would not approve of.
Secondly, "Does he still have feelings for his ex" -From what you wrote, I can confidently say that he does have some feelings. As to the extent of those feelings, no one will ever know but him.
For some people, it's hard to let go. Think about it...You had an intimate relationship with someone for x amount of years or months, shared some times only you both care about, and then it ends. It's like a kind of "death" and depending on how long ago they broke up, he is keeping the fire somewhat ignited, to ease the mourning process. This may be his way of "healing".
OR, he could feel guilty about how his last relationship ended, and is coming to terms with it in his own way as to not feel like he is the bad guy.
I'm not sure of the whole situation here, but I'm trying to understand why he would do something like that and give him the benefit of the doubt, because after all- he is your husband.
However, This is a disrespectful thing he is doing to you.