Hello!
I'm using GrapheneOS on a Pixel 9a for a bit over a month and it's unique (in a good way) compared to my previous experience on Android (with Samsung and Huawei phones). I've never owned an iPhone, so unfortunately I can't make a comparison for you.
Battery life has been great. I'm not sure what you mean by sandboxed GPS (as far as I am aware, there isn't any extra sandboxing going on for that), but there's no reason it would eat up any more battery than any other Android phone and Google's Location Services are still available as an option if you prefer.
Play Integrity is mostly a blocker on banking-related apps and wireless payment apps. You may also find issues with government apps depending on your country. I haven't had any issues so far. It's best to search the forum for any apps important to you to make sure no issues have been reported. Here's a list of some banking apps and their compatibility with GrapheneOS: https://privsec.dev/posts/android/banking-applications-compatibility-with-grapheneos/. Most apps have no reason to restrict you and there will nearly always be some alternative for those that do. You can also contact the developers to fix the issue, it's mostly done out of ignorance and the assumption that all custom Android OSes are insecure.
I can't answer the questions regarding governance, your best shot is probably the FAQ page or hopefully a developer answering your question here.
For a keyboard I'm using FUTO keyboard, although I really don't like them as an organization. Floris board, Heliboard and Gboard are the most popular options and great alternatives to whatever you're used to. Unless you need a really niche feature, don't worry about the keyboard. You can use what you already love by downloading the app from the Play Store.
The standard camera app is fine for taking photos and videos (the QR scanner mode is also really handy, for me at least). Extra modes like slow-motion, night mode, etc. are best in the Pixel Camera app which is really nice actually. I rarely need it though, I'm not that much of a photographer. From what I can tell, Google Photos isn't really used as a gallery app, there are many great FOSS replacements. Aves (available on A crescent), Gallery (yes it's really called that) and my personal favorite Lavender Photos are all great alternatives. I also use Immich for backing up my images and syncing.
For browsing, I use both Vanadium and IronFox (slightly hardened Firefox fork, available on Accrescent). Vanadium is the only recommended browser and it's security is unmatched by any other browser. I accept the security tradeoff with IronFox because I can get much better ad blocking, extensions and features like clearing browsing data on exit and incognito mode by default. Vanadium still ends up being better for your privacy and security though.
I can't really recommend a DNS service, for the best privacy you should use a reputable VPN with their default DNS server to prevent leaks. Some also block ad and tracking domains for you. The private DNS setting, as far as I can tell from the FAQ is not recommended, it's a system-wide setting and can make things weird. I'm self-hosting a VPN with my own DNS server, mostly so I have control and I can access my home network.
I can't answer about the Mastodon account, I don't follow it at all. The accusations are part of a constant battle from certain individuals wanting to harm the project's reputation. I wouldn't pay too much attention to those trolls.
I'm always looking for ways to improve my privacy and my phone desperately needed an upgrade (it was around 6 years old and hadn't received updates beyond Android 12), so I weighed my options and settled on GrapheneOS as an option where I have much more control than any other mobile OS. It's a breath of fresh air to actually manage your phone alone without the manufacturer shoving AI and bloatware into your face and tracking your every move with invasive system apps. You can use all that if you want to, but the point is that it's completely optional. The 7 years of updates also slightly justified the price of the phone for me. So far I'm really happy.
I don't explicitly back up my contacts, I rarely do manual backups but I did set up app and file backups in the settings and I'm fairly sure that backs contacts up too. I'm sure others have better solutions, probably synced with a CalDAV server.
GrapheneOS is based directly on top of AOSP and inherits both its advantages and its drawbacks. This means that many quality of life features you may find in other OSes or Pixel OS are not available, because they're proprietary. For example, I've been looking for a parental controls solution (available on Pixel OS and all my previous phones as a built-in feature, yet there is no such thing on GrapheneOS). Many features will also get overlooked because the developers have limited time and are mostly focused on improving security and I totally understand that. It's a small compromise you'll have to make. At least we finally got lockscreen widgets! 😁
I hope that (mostly) covers you, feel free to ask any other questions you may have!
EDIT: While I was writing this someone with more info about the project status answered, so I think all your questions are answered now!