The Death Valley forum https://dv.netllama.us/dv/
would be helpful here. Also the National Park web site https://www.nps.gov/deva/index.htm
1. From Lone Pine to Furnace Creek (park headquarters) is a two hour drive. Aside from staying in the park, the only closer city is Beatty, Nevada. Beatty is not convenient for Mt. Whitney. Inside the park there is the motel at Stovepipe Wells, the Furnace Creek Ranch, and the Furnace Creek Inn ($$$).
2. For a day trip consider these places. A web search will provide plenty of information.
Badwater (lowest point in the western hemisphere)
Twenty Mule Team Canyon
Mesquite Flat Sand Dunes east of Stovepipe Wells
Devils Golf Course
The Racetrack (home to moving rocks)
Titus Canyon (drive through)
Visitors Center at Furnace Creek
Historic Furnace Creek Inn (sit on the patio and drink a glass of wine while watching the sun set behind the Panamints)
If you have a few days for hikes, these are worth doing. A web search and guidebooks will have route information.
Furnace (or Funeral) Slot Canyon
Moonlight Canyon to the Moonlight Natural Bridge
Note that virtually all hikes in Death Valley are cross-country. Navigation skills are required. A topo map and GPS are very useful.
3. How many days? How about how many weeks? Seriously, some great highlights can be seen in a day. You could also spend a month and still have more to see.
Pay close attention to the weather. DV is not a good place to visit in the summer when temps reach 100 to 120 regularly. Humidity is very low year round. Carry and drink plenty of water and sports drinks.
Unfortunately, two terrific web sites Death-Valley.net and panamintcity.com were recently closed by their operators. There is still a shelf of guidebooks available. The best guidebook is Hiking Death Valley: A Guide to Its Natural Wonders and Mining Past by Michel Digonnet.