1. Google has a tool call Tag assistant can help to detect all the tags on web pages. Does DTM has similar tool can tell me which tags on my web sites?
DTM does not have something like this.
2. When migrate google tags into DTM, where should I put all the codes? Should I place them in third party code window?
You can add Google Analytics or Universal Analytics as a Tool or as a 3rd party tag.
3. What will be the best way to manage all the tags in DTM? or move from Google tag manager to DTM?
It's a manual process, one tag at a time. How difficult it will be depends on how your tags are currently setup in GTM.
Additional Notes : If you are willing to hold out for a bit, Adobe has a new tag manager called Launch, which is supposed to be a lot better than DTM. In particular, it is supposed to have an API for programatically managing tags, so you should be able to write a server-side script to automate migrating tags from GTM to Launch (though TBF, the time and effort in creating that script may be more than just doing it manually. But maybe someone else will beat you to it and share!). Anyways, it officially publicly released a couple weeks ago but isn't actually available to the general public yet; supposedly within the next couple of months it should be, though.
1) We leave tag auditing to others in the industry that already have great solutions. Many of our customers use Observe Point to do this, but you're welcome to use whatever tool you'd like.
2) It depends on the tag. If we have a tool built in DTM for that tag like GA or GUA, I'd encourage using the tool. Otherwise, you can use the thrid-party section.
3) It would be a manual process to migrate from GTM to DTM.
As the previous reply mentioned, we have released our next generation tag management platform called Launch, by Adobe. It's a rolling release so you need to request access and wait for your company to be selected via lottery. You can request access using the link at the top of the documentation here. As indicated above, Launch is a platform so it was designed with an API first approach. Eventually, developers will be able to write their own extensions for the Launch platform. I could very easily see someone writing something to help with migrations from other vendors.
If you're interested in learning more about Launch, feel free to visit Adobe.com/go/launchme to view or register for one of our demos