I feel for you as I've been here but I'm afraid you're going to struggle to get anywhere with the mother because it's her child and not yours. This doesn't mean to say it's impossible. 5 years of having a step-son myself and I'm often told by my girlfriend not to tell off her son or be too harsh when I believe I am being reasonable. My step-son is now 7. Her excuse has always been he's just a child and whilst I agree with this on the whole I feel it's not right to let bad behaviour go unnoticed.
After 5 years the thing I've learnt the most is to let the mother do the discipline and for you to back her up. After all it's her child and it's down to her alone on how she wants to raise the child. It's a very difficult situation to be in because you feel they are being naughty and you feel you have to act. I still struggle now when I see my step-son misbehaving. I'll tell him off and 2 seconds later his mother is in asking what's going on? so let her do the disciplining and be there to back her up.
Sit down with your wife and explain to her why you feel it's good to discipline. You don't need to always be on their case to discipline. You just have to set out guidelines the children must follow. Agree these guidelines together and together uphold them.
The things I very much push for with my step-son is similar to you. Manners and to be respectful. This is something that was drilled into me as a child.
Manners. This tactic worked very well with me and my step-son. When they ask for something and forget to say please. Prompt them. Something I say is what's the magic word? or as I've now been doing it for a long time I give him a look which prompts him (raised eyebrows and very slow moving until he says please then I quickly move so he knows). When you hand them something, don't let go until they say thank you. Make sure to prompt them if they are struggling. Don't just hold it and snatch it back. It's about being subtle. By doing this you are helping them understand when it's right to say please and thank you. Other manners such as excuse me can be prompted should they pass wind etc. You must continue to do this for a long period of time. It's not going to sink in overnight and it may even take years.
Respect comes from you. They look up to their parents (including their step-parents) so act respectful and they will start to copy. Opening the door for the mother is a simple act of respect that they will pick up on. Don't argue in front of the child. Make sure you are always saying please and thank you. It's all down to you this. Children love to mimic so make sure to always be respectful yourself.
I would like to add that our relationship (mother and I) is very good on the whole. We are both working parents, with two children (one who I'm step-father too). We have had our ups and downs and the biggest reason we've argued has been because of my step-son. His father is very much involved with him so I've now decided to take a cooler approach and let the mother handle most of the discipline unless of course he is completely out of order. The biggest thing is communication. If I feel I am in the right for disciplining my step-son I will talk to the mother and explain why I told him off. I do this out of sight of the child and when we are both in a calm manner. There is nothing worse than arguing why in front of the child and having it escalate.
As I said above, communicate with the mother. Express you wishes and talk through any issues you encountered through the day. Get her opinion on how to handle the situation. Like I said at the top it is down to her how she wants to raise her child.
A bit of personal advice. Try not to take it too seriously. What I mean by this is don't let it get you down when the mother overturns your decision. Instead take note and talk to her about it later. Whatever you do, you don't need to give up. Children grow up and as they grow they come to understand more. By giving up you're not allowing for your hard work of being a step-parent to show. I've had some real issues myself becoming a step-parent and as I've said in another answer it's harder to be a step-father than a father and that is no exaggeration.
As for your younger son picking up the behaviour of your step-son I wouldn't worry too much just yet. Just make sure you don't show favouritism and ensure both are equally disciplined. Good luck and I really do hope you can stick with it.