Dear Uroš,
1. Where to stay...
I stayed two times in hotel Odesa which is on the way to Odesa, 10 min drive from the Center, on the left side. Very good hotel with good service, food and nice staff. Their parking is guarded 24h/day, there is no need to worry about safety if you stay there, very professional guards armed and nasty looking as they should be.
I recommend booking in advance because you will get much better deal that way, also pretty much all hotels in Ukraine have guarded pakings so don't worry about that.
2. Which way to take from Hungary to Kiev...
Enter in Ukraine at Chop and just follow the signs to L'viv and Kiev. It goes like this Chop-Stry-L'viv-Rivne-Žitomir-Kiev. Also you can take road road L'viv-Ternopil-Hmeljinski-Žitomir-Kiev but I would rather go with the first version. In Ukraine there are no autobahn's as in the EU, first of all you don't have to pay second every few to ten km you can make 180 turn on the "autobahn" very dangerous! Drive carefully. The way from Hungary to Kiev is mostly magistral road, but their roads are really wide and many times I have seen two trucks and a car in the middle all in two lanes... So you get the picture
3. How long...
On the first day you need to go across all of Hungary, I recommend not to push it all the way to L'viv if you go to Ukraine for the first time, keep in mind that you will lose some time on the border, they like paper stuff.
When you enter Ukraine you will find yourself in Zakarpatska oblast, there (some 80-100km) are alot of really good restaurants and hotels there. You will see when you start to go uphill.
On the second day you can go and visit L'viv, it is a nice place.
From L'viv to Kiev if you start in the morning you will come to Kiev at evening. It is a long drive, but it is like that in Ukraine.
4. Tent...
Well on your place I would leave the tent at home. Stay out of trouble...
5. What to see...
In short, forget about Chernobil. L'viv, Kiev, and Krim are places to see.
6. Etc...
Drive cerfully. I have seen some of the worst accidents in Ukraine. Ukranian police is much much better than the Russian, but keep within legal limits. Main roads are generaly ok, but local roads... you don't want to go there
If it is raining avoid little ponds because the hole might be deadly deep.
Keep it safe
Nikola