About daily maxing, one of the biggest benefits is that you get used to the feeling of heavy weights and you also get a lot of practice on your technique - it's easy to have good technique with light weights but it's hard to hold things together when things get heavy. I mostly did that because I was having issues with my squat, but it was good for bench as well. Like the other guy said it's important to listen to your body, I was doing a lot of down sets the first week and it was getting to me. There isn't a lot of information out there on it, but there is an e-book by Matt Perryman called Squat Every Day that explains a lot. For the record, I was training 6 days a week and just doing some squats with the bar on my day off to stay loose (I train in my basement).
When I started I wasn't sure if I was going to do it for just a training cycle or longer, my plan was to just see how things go and get some practice on technique. Unfortunately after 6 weeks I caught the flu and for a couple days I was not really able to train, I deloaded for a week and then tried to start again, I managed 2 workouts but I just didn't have the energy or motivation to continue. Either way it was worth a try and my squat technique improved a lot. You aren't necessarily going for a 1rm or PR every day, just the heaviest squat and bench you can manage with good technique and then a couple down sets. For deadlift, at first I followed John Broz's suggestion of doing speed pulls a few times a week but I found that more tiring than maxing out. Mainly I did the comp. lifts but I also used variations, for squat: SSB and front squat, paused down sets, bench: close grip, floor press, bands, slingshot, for deadlift the only variation I used is deficit pulls. On day I wasn't pulling I would do some work for lats and abs.
So after that I went back to my own RTS-based program, I noticed that weights that seemed heavy felt really light now but doing high reps (by that I mean more than 3) for multiple sets felt like something brand new and I was getting sore as hell. Around the same time there was a discussion on the RTS forum on daily training and Greg Nuckols said he uses daily max cycles as a transition block and to increase volume responsiveness, the idea is that otherwise you would have to continually increase volume to make gains as you get stronger and you work capacity increases. He has an article on this site - I think the title is something like "Bulgarian Training" - and the main difference with the way he does things is that he uses no down sets at all, only singles. I plan to go back to that style of training at some point in the future, but I would take a slightly different approach. John Broz says "how you feel is a lie", which I agree with to a certain point but if you are just constantly accumulating fatigue and doing lots of volume there is a good chance you are past the point of diminishing returns. In case you are wondering, I don't take steroids. That seems to be a common presumption when people hear about daily training.
Anyway, from my own experiences and things I have learned along the way, all lifts can be trained several times a week as long as you don't do too much volume in a single session. Some people have this idea that beginners need a 5x5 program but once they pass that stage it's better to use less frequency for each lift and higher volume per session, there is no doubt that some people make good gains like that but for the vast majority it's not the way to go. I'm not recommending daily maxing to you or anyone, but it's one approach to training that can work if done properly.
As far as exercise selection, bench press and any variations that help your weaknesses are the way to go. There is no magic formula, just work on sticking points and build strength off your chest (assuming you are a raw lifter). I do military press here and there but no other shoulder work and I do absolutely no tricep isolation work.