Is there any problem with installing the GTM and DTM in the same site? I searched and read about being careful with Data Layer, but I'd like to know if there's any conflict between the tools
I second the notion that having more than 1 TMS on a site is not generally a good idea. However if you are forced down this path there are a few things to make sure you are doing / not doing.
1) NEVER EVER EVER use one TMS to deploy another TMS. It's bad mojo.
2) The success rate of having code in one TMS try to grab variables that are being set/used in the other TMS will be quite low and it most likely not something any TMS vendor will support.
There might be other items but these are the 2 that stand out the most to me.