Madi100,
My 2cents, I'm downtown every day of the week - walking from the south end of the loop (near the board of trade) all the way to the Hancock building (north end of Michigan avenue).
1. Palmer House - is it old - yes, it's an older building, I've been there recently for some seminars - from people I've talked to you'll be fine. The area it is in is in the loop, south of the river, which is not the heavily touristy area. That being said there is a lot down that way - you are right by a ton of theaters - the Schubert, Oriental, Goodman, Chicago - check www.chicagoplays.com or www.chicago.il.org (convention and tourist site for tons of info). Safety is the same down there as it is on the north part of the river - just be aware of your surroundings.
2. Parking - always tough in the city, Grant Park underground on Michigan avenue by your hotel would probably have an overnight rate of $19 - but you'd be lucky to find anything overnight cheaper than $15 within a 8 block radius of your hotel.
3. The website above will fill you in on the free trolley, there is a stop by the Palmer house and the trolley runs by alot of main places (Water tower, Sears tower, Navy Pier,) but it doesn't run 24 hours. I walk alot so walking doesn't bother me but cabs are fine also. The water taxi runs on the river from the near west side (near Sears tower) to over near Michigan Avenue. That is also where you can sign up for a Wendella boat tour - come as you are and take the city/lake tour.
4. Your right after the blues fest and just before Taste of Chicago - but there will be plenty to do. Have Chicago Tourism send you out a packet and check their site. There will be theater, baseball, shopping, museums, etc.... If you want suggestions on lunch/dinner areas let me know - there are great places on Rush/Oak area on afternoons to sit outdoors have a drink, appetizer and relax.
Let me know if you have any specifics after checking everything out, I've been to way to many restaurants, clubs, theaters, etc.. down here.
Redbirgrad