Dedicated server hosting is a type of web hosting plan where you rent an actual server (a rack-mounted, internet-connected computer).

Probably not, but maybe.
Most people who need a highly configurable web hosting server with full root access are better off with a VPS hosting plan. They are usually less expensive, and — if it is a cloud-based VPS plan — easier to scale.
The use-cases that specifically for a dedicated server plan, as opposed to a VPS plan, are increasingly rare — and if you have one, you probably know about it already.
The most likely reasons to need a dedicated server plan have to do with security and regulatory compliance — VPS plans may not satisfy certain types of privacy laws, such as HIPAA.

Only if you need it.
Most of the advantages traditionally gained from dedicated server plans can be had today from high-end VPS plans.
The only reason to use a dedicated server is if you have a use case that requires control over the server hardware and/or personal knowledge of the hardware's physical state and location.

Yes. But you probably don't need to.
If the only thing you are trying to do is set up a public-facing website, you are more likely to need a VPS hosting plan.

If you are just trying to setup your own website or web app, and need web hosting, then — no, you almost certainly do not need colocation hosting.
You might benefit from colocation hosting if you currently own your own server hardware, and want to store it off-premises for reasons of security, economics, or faster connection to the internet. You may also want to look into colocation hosting if you need to purchase your own server equipment in the future, but don't want to (or cannot) provide your own network infrastructure and security at your present location.
You may also wish to look into dedicated server hosting, in which you rent both the hardware and the datacenter space.

If you are looking for colocation hosting, you can use out hosting features search tool.

Managed hosting is a type of web hosting service that combines either a shared hosting or VPS hosting platform with a high level of customized support and management.
Managed hosting takes a number of different forms, so it's hard to make generalizations about it. But it usually involves some kind of active support for your installed applications, security monitoring, automated backups, software upgrades, and server configuration tweaks.
Some managed hosting providers have specialized services for one or another specific applications, such as WordPress. These providers have usually designed their server setup and support structure around the app, so they are able to provide a high level of satisfaction to a very narrow set of users.

That depends. The average website owner doesn't, but some types of users can really benefit from it.
If you are a medium-to-large business that runs depends on a web-based application (such as an online store or an internal Sharepoint server), but you don't have an in-house SysAdmin, you probably would do well to have a managed hosting provider to help ensure a high level of uptime. This pretty much is the case with any kind of mission critical server.
Additionally, highly trafficked content websites or blog networks can usually see both an increase in performance and a decrease in administrative hassle by switching to Managed WordPress hosting.

WP Engine is a Manage WordPress Hosting company. This means they provide web hosting only for sites built using the WordPress software. They provide a high level of support, as well as a server environment specifically optimized to work well with WordPress. They handle installation and upgrades, and a number of routine and specialized maintenance tasks.

If you run a very high traffic WordPress site, WP Engine might be a great hosting choice for you.

For mission critical WordPress sites with a lot of visitors — such as commercial blogs, multisite networks, and high-traffic content businesses, it probably makes sense. As compared to hosting with a non-managed VPS or shared hosting, you'll get faster load times (especially under load), fewer tech problems, and a lot less time spent on site management.
For personal blogs and small business sites, it probably isn't worth it.