Winter trekking in High Tatra, what about accommodation?

Accommodation

Yes, there are some accommodation options in the Tatras mountains. I do not know precisely where you want to go but you can check on Open Street Map if there are shelters on your way.

According to the Wikivoyage page for the Slovakian High Tatras, there are many huts but they commonly close for the winter. The linked website has a list of Chalets. You have to click on each link and then switch to English version (unless you read Slovak) to figure where it is and if it is open in winter.

Avalanches

Regarding the safety, as it seems pretty snowy, it is probably mandatory to be correctly equipped for any snow slides and avalanches. Maybe you have some resources from your local/national hiking association about that topic, the Polish Tatras national park website links some website on the topic (it is translated so not the best). There is a Wikipedia page on the topic.

Trails information

Also, many trails are closed for the winter. As there are no trails marked as open for trekking and hiking, I have no idea if they are all closed. Some are open to alpine and cross-country skiing.

I think these resources are a good starting point. You can find more information on the website of the Polish National park of the High Tatras. In particular they have reports on avalanche risks.

I would recommend you to browse these resources and search on the web for more (unfortunately most resources are in Polish or Slovak, but you can use translators like Google translate). It is very important to plan ahead your itinerary, to have a paper map (especially as in cold weather, the batteries of devices tend to last a shorter duration). According to these resources, winter trekking with stops in shelters are for experienced trekkers. Please be aware of the risks.