Do I need permission to publish a mobile application for a website I do not own?

You mentioned that "I use the name and logo to draw attention to people who want the mobile app." Since that is not your name and not your logo, you are not permitted to use them for your own commercial purposes. Trademark and Unfair Competition laws prohibit the exact thing that you are trying to do (ie use other's good name for your commercial gain). As far as the DMCA take down notice is concerned, that is a mild remedy compared to the availability of trademark infringement claims that may include attorney's fees and triple damages. Also, since the intellectual property owner is in the UK, you may face an international lawsuit that you cannot defend against and then you will have a civil judgment on your credit report. You are best to abandon your business idea and move on to something that does not violate other people's intellectual property rights. You should consult a lawyer on this matter.

Yes, it is illegal to use anothers product, name, and/or logo in any manner without their express permission or to imply that you have their permission when you don't. The location of the website is irrelevant. Taking something that doesn't belong to you is theft.

Permission is likely required, for all of the reasons set forth in the Question. This is especially true given the threat from the website owner. See if you can work out a deal with the website owner, a "win-win" is better than making lawyers rich.