A search on CMS yeields thousands of results. What is the most popular (& good) CMS these days? specifically a CMS that supports mobile devices and works well with a website that includes lots of content & apps.
YMMV, but the best solution I've found for this is either:
1) If your CMS just needs to be used in-house, then roll your own. It takes a week or less, you can deep-integrate it into your applications, and you'll be guaranteed to get more or less exactly what you want.
2) If you either need it to be exposed to external/non-technical people or you absolutely do not want to spend time developing CMS features, use WordPress for content pages and your app stack of choice for app pages. You can have them share the same domain if a requirement for e.g. SEO by using reverse proxy tricks -- trivial to Google if you know the words "reverse proxy." If not a requirement, don't do it that way.
One of the nice parts about using WordPress is if you have a high-generalizability problem like "It needs to work on an iPhone" then you are basically guaranteed that some 17 year old designer has hacked together a plugin which makes it mostly work on your iPhone. The downside of using WordPress is that... feel free to rephrase the last sentence.
I've actually wondered about this a bit before. You're kind of the "Marketing Engineer" around here so I'll ask you.
Do you do what you described in option 1 or 2? I know you use WP for your blog but what do you do for the marketing/landing pages of your apps? Do the marketing pages live in the same code base as the app? I'm inclined to do it this way personally so that integration of something like ABingo! is easier.
Bingo Card Creator is 95% #1, Appointment Reminder is #2. Client sites depend on what I can convince them to work with, largely tending to follow whomever "owns" the marketing site internally (engineers prefer #1, non-technical folks prefer #2).
There are likely to be pretty reasonable Wordpress themes and plugins for most common use cases, though. For something responsive, really the default twentytwelve theme or Responsive I think are good bets. Every now and then though you come across something weird and have to write something custom for the functions file and you end up hating life because Wordpress has some byzantine crapfest way of doing things which should be simple (walkers, i'm looking at you.)
The real downside is security. The hardening wordpress[1] article is pretty good but I also found some decent tricks in the .htaccess file for the html5 boilerplate[2] that seem useful too. You end up having to install it, then fix it, which is never good.
There's a lot of factors here. Use the right tool for the right job.
If you want to put in the for to doing some learning Drupal is a good. Its got a massive community. Its not as "plugin" friendly as WordPress and its more server side Dev friendly than WordPress. For my personal use I use Mezzanine which is a Django project. I picked it because its got a lot of solid basics that id need, is python based yet has the backing of Django community.
Unless you roll your own you're always going to become frustrated with a decision that someone else has made for yourself.
It has one of the largest ecosystems (if not the largest), and you can usually find multiple plugins to do whatever you want. You can self host, use their free hosting, or pay for their premium hosting. It has a variety of responsive themes (both free and paid), and a mobile switcher that will auto-detect if you are surfing from a mobile or desktop device. Due to its popularity there's tons of tutorials and help on bunches of forums. It may not be the best if you are looking for a particular feature set, but it is probably the best all-around solution.
I'm actually doing this now. We're using jekyll for a tablet magazine site with several non technical staff using github and prose to manage the content.
It's actually going fairly well. There have definitely been some issues, but it's been great.
I've used mezzanine before but ultimately removed it from the code. If you're interested in adding a company blog to your app's domain, it just touches too many things and makes a mess. I use zinnia now http://django-blog-zinnia.com/.
I think mezzanine is a great alternative to wordpress. It makes a great blog if that's all you want. If what you need is a CMS to handle the content side of your app, you may want something a little more flexible/light weight.
RE: Part I get it... It's kind of just like Wordpress, great for blogs and everything else is difficult IMO. The only difference being you can still use Django if you need to. I guess... I don't know enough, just enough to be dangerous. :-)
I use Drupal. As others have mentioned, these have their downsides but on the upside, it's got a decent developer community and integrates well with the app itself. (usehuman.com)
1) If your CMS just needs to be used in-house, then roll your own. It takes a week or less, you can deep-integrate it into your applications, and you'll be guaranteed to get more or less exactly what you want.
2) If you either need it to be exposed to external/non-technical people or you absolutely do not want to spend time developing CMS features, use WordPress for content pages and your app stack of choice for app pages. You can have them share the same domain if a requirement for e.g. SEO by using reverse proxy tricks -- trivial to Google if you know the words "reverse proxy." If not a requirement, don't do it that way.
One of the nice parts about using WordPress is if you have a high-generalizability problem like "It needs to work on an iPhone" then you are basically guaranteed that some 17 year old designer has hacked together a plugin which makes it mostly work on your iPhone. The downside of using WordPress is that... feel free to rephrase the last sentence.