I do think if he has time to go out, then he should be supporting you and helping out. Maybe if you're breastfeeding or don't feel ready to leave the baby yet, then a night out is off the cards for now, but make sure he knows you intend to have one at some point down the line! And as others have said, he can always help by looking after the baby while you sleep.
As for him being on the computer rather than interacting directly with the baby, I think this is fine, as long as the baby is happy and has all her needs met. For example it would not be okay if she had a dirty nappy and he was happily ignoring it, playing a game or something. I don't think babies need that much interaction at this stage - the only reason we tend to do so much with the first is that we don't have any other children to attend to. However, it would be nice if he did some other things with her, like bathing her (A great one is if he gets in the bath with her - the skin to skin is excellent and babies seem to like it more. DS used to scream if bathed alone!) or carrying her in a sling or something. Or you could do things as a family - going swimming, to the park/beach (perhaps better for a sunnier day!) taking her around the shops, stopping for a coffee, visiting friends/family etc.
I think you need to discuss the dummy thing and decide between you when it will be used and he needs to stick to that, unless he has literally tried everything else.
There are a few things sticking out in your posts to me - the fact you do all the housework and cooking, the fact when you discuss things he agrees but then does the opposite, the fact he gets a night off without even offering or mentioning to give you a night off in return, whether you go out or not, and the bit that you said in your OP that he is unsupportive to the point that you ended up crying and ranting at him for it. Yeah, he might have a different parenting style to you/feel out of his depth and feel it's okay to opt out because society enables fathers to do this, but alternatively, he might just be being a selfish, unsupportive dick. If this is part of a wider pattern of behaviour or he was like this even before the baby was born, then the second one is probably more likely. If that is the case thn sadly I think your efforts will be wasted trying to get him to understand.