1. Regardless of who does the initiating, in my mind, going to someone's house to watch a movie or eat dinner is NOT a date. It is fine for people in relationships to do this -- when you reach the point in your relationship in which you are exclusive, boyfriend/girlfriend -- then spending time at home is cool. But notice even when married couples (like the Obamas) talk about having date nights, they are NOT talking about popping a DVD into the machine or having dinner at home. They are talking about going OUT.
As far as I'm concerned, I might invite a man in for a bit after he's taken me out on a date, but until he's my boyfriend, we are NOT getting together for a stay-at-home "date."
Now, as you know, I'm pro-courtship, so my dating advice is very purpose-driven. If people want to chill at home all the time, that's fine for them, but I feel that is a privilege given to someone who has already proven that he is looking for more than just someone cool to go out with here and there and that he is pursuing me for a long-term relationship that leads to marriage. Therefore, I turn down all suggestions from men that involve coming over for dinner and a movie and never dare invite a man over for such... plus, one needs to think of safety considerations too. You don't know this man like that.
2. I don't "woo" back. In the courtship process, that's his job, and my role is to be receptive and encouraging. Until we're officially in a relationship, I'm a free agent, and he needs to keep coming at me to get me off the market. Now, at the end of a date, I often say, "That was a lot of fun. Thanks for inviting me to do blah blah blah. Let's do it again sometime." Then I sit back and if he doesn't ask me out again, I move on.
For me, I expect him to keep wooing until we are exclusive. Women don't "woo."
The men who are interested don't need me to ask them out or do a thing. They keep suggesting and planning future dates and I keep accepting. Shoot, the good ones ask you that same night... or very soon after.
If you have to start thinking too much about suggesting the next date, you need to fall back.