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You can get 4k at 60Hz with DP but you will need to use straight connection in order to do that, Intel do not recommend any adapters. For better performance straight connection is the best option.
If your TV does not have display port, you can try installing the INF file for your TV, you will need to download it from your TV manufacturer website.
You can also try uninstalling the old driver and install the generic driver from the Intel website, you can download it here:
Not yet. I can't get an INF file from the TV manufacturer because they don't have one. I tried to make a custom one using the instructions from Microsoft but that's not working either. Lastly, I have to run an DP to HDMI adapter because the TV doesn't have a displayport of its own. So, for now I am living with the issue because I think it may be a deficiency in the laptop graphics card.
With an active adapter from DP to HDMI you might be able to get better resolution, also I suggest you to contact your laptop manufacturer for them to give better specifications about the laptop as they are more familiar with your component and the parts interact with each other.