I’ve found some photos of the chalet. Appears there are blankets,bed linings, pillows in rooms.

Morskie Oko definitely yes. Czarny Staw not necessarily, the trail will probably be covered in snow so risk of slipping might be high. Of course it's worth checking out the beginning of said trail to see how the conditions at the given time really look. Please be warned that due to public holidays, Morskie Oko can be expected to be extremely crowded on Apr 27-28 and May 1-5, so if you fall on those days it's a better idea to choose another destination.

зависит от сезона. мы поднимались в июне и погода была не самая лучшая. важно, чтобы были удобные трекинговые ботинки, остальное опционально. готовьтесь к тому, что в одном месте придется карабкаться по почти отвесному камню, держась за цепи.

It is possible but from polish side it takes about 13 h to go up and down. At least for calm and slow walker. There is a shelter after first 10 km to stay for a night but book a room in advance. It is well marked so no need for guide

Starting 4 am would be more than enough. Just a tip- there’s also a shelter in slovakkia 1 h before strbske pleso. Similar to Morkie oko. But weather yes is a risk. Check mountain weather forecast/ it’s pretty accurate as far as possible. Maybe you can find weather window in july.

I know this is too late for you, but in case any future travellers read your question, the moment from which onward you can climb Rysy from the Polish side without winter hiking gear like crampons or ice axe usually comes between June 20th and July 1st (with higher probability of the latter). From the Slovak side it usually comes around a week earlier than Polish, due to it being the southern slope. The Slovak trail (like all higher-altitude trails in Slovak Tatras) is only being opened on June 16 and in the warmer years you might be already able to hike Rysy without crampons as early as this, but it's not a given.

No, the trail onto Rysy from the Slovak side is being opened (like almost all the trails in Slovak Tatras) on June 16 and I guess in most years there can be little enough snow to go safely around that day. From the Polish side it is allowed to go whole year, but as a rule of thumb the very last days of June is when you don't need winter equipment to ascend Rysy (earlier for other trails, Rysy are one of the last to melt usually).