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.