As far as homebrew stuff goes, there really isn't a lot working online. You already know about the Doom and Quake ports, and Quake World is still in the works and not even publicly released yet. Shuoma is still working on getting the other former online games resurrected first.
I really don't see the insistence on using something else other than the official controller for FPS games? Just think of it like how it is on a PC with keyboard, W, A, S, D movement and A, B, X, Y, are practically the same 100% digital. Sure lack of an extra analog is disappointing, but the analog stick is used for free look in most games, and can be used in all FPS games on the console.
Sure, mouse is superior, and I am primarily a PC gamer too at least as far as for modern games. So I get the whole argument about using mouse and keyboard on the DC. But as far as working PS2 laser mice, it is going to be difficult to maybe find those now. There are those PS2 to Dreamcast mouse and keyboard adapters you mentioned, as well as PS2 to USB adapters. Not sure if those would actually work on the DC. I heard people say that had problems with them.
Just practice getting used to playing Half-Life. It really has similar controls to Goldeneye 007 on N64 when you think about it, due to the controller layout. Strafing is done with the face buttons which are digital C buttons on that system. PSP FPS games were handled the same way as well for that matter. So it is not like that control scheme is unprecedented or anything, at least 3 systems worked that way.
It just takes a little time getting used to it but once you get the hang of it you stop thinking about it and it just comes second nature.
Also, don;t forget, Retro-Bit will eventually release wireless dual analog controllers for DC that are officially licensed by Sega, which will have Saturn Styled 8-Way D-Pad, and 6 face buttons. It is our hope in the community that games could be hex edited to use more modern layout on these controllers since the Maple protocol actually supports up to 2 analog and 2 D-Pad inputs at once. So eventually this should all be taken care of, you would just have to use copies of games that were prepatched for the new layout.
Edit: You can see my thread here about the new Retro-Bit controller. I really have been super excited for this since they revealed they were making all new controllers at CES 18'.
Since my Retro-Bit Sega controller megathread is about to die over with Assembler, I just wanted to continue the thread over here. They have shown off the prototype of the new Dreamcast controller which will have a hybrid design of the Dreamcast controller with a PS4 thumbstick, the Saturn...
www.obscuregamers.com
I also update the thread whenever more news is revealed. It has just been a while since we heard anything about them. They told me on Twitter that they were still pending approval by Sega on the final design.