Please help with my first trip in DC and NY

How hard and expensive to find parking during daytime when I visit the sites above in DC? And where?

Don't bother. Take transit from Alexandria into DC.

Tours By Foot

Why so many tours? DC is easy to explore on your own.
For someone who likes museums, you should be devoting those two days to the various Smithsonian museums. Yet they aren't even on your list. The Smithsonian is the largest museum complex in the world.

First day - Alexandria Old Town

Old Town is nice, but it's small. Most of it is shops and restaurants. You could easily stroll it in an afternoon or evening. Spend more time in DC, then come back a little early one day for this.

Capitol Hill & Library of Congress

Book an official Capitol Tour instead.
And just walk to the Library of Congress. There are free tours of the Jefferson Building (definitely worth it).

How hard and expensive to find parking during daytime when I visit the sites above in DC?

Very, and very. And let's not forget traffic. I have relatives from out of the area, and they were surprised when we hit traffic at 10 am. Another time, it was 2 pm. They had trouble understanding that this area is traffic-choked. My usual response for "why is there so much traffic," even on weekends--"it's DC."

So I'll repeat the suggestion above--take Metro.
(But be aware that they are currently undergoing SafeTrack, an intensive program of repairs, which may shut down various sections. See info here

And probably visit Georgetown for dinner

Georgetown is fine for a stroll, but the chains are taking over. You may prefer to head across the bridge into Rosslyn-Ballston corridor area of Arlington for dinner.

Day 4 - Free Tours By Foot (NY in one day at 10am to 4pm)

Again, I think you're taking too many tours.
NYC is also easy to get around on your own.

Also, consider the Lower East Side Tenement Museum.

Day 3 - Statue Liberty with Pedestal at 12pm

Book in advance. A week or so.

Is there any pick-up service to/from airports?

Transportation options.

http://www.freetoursbyfoot.com
I think it's helpful since it's my first time traveling to DC and NY with my teenager sister. Thanks for your inputs.

It's fine to do a tour, but to fill your days with them is rather meaningless. You will tour everything yet see nothing.
Scale it back and explore the cities on your own. And again, some of those places you "tour" really deserve more time than what the tour will give. Not setting foot in a Smithsonian museum is a travesty, and some could easily take a half day or more to enjoy.

Staying near Prospect Park can mean various neighborhoods. Park Slope is totally safe. Lefferts Gardens and Crown Heights east of the park are slightly less safe, but you should be OK, no problems if you are careful, don't walk alone, etc.

From this neighborhood, you can just use the subway system to go to JFK airport. It will take 1/2 hour to 45 minutes, and you'll need to switch to the airport train at the Howard beach stop on the A train.

Thanks. I'll be near the Planet Fitness on Flatbush ave.

I'm a big fan of tours, but agree that you don't need them to see the sights of DC - better to explore on your own than be rushed by a group and not get to linger where you want. That said, I loved the Monuments by Moonlight tour when I visited - great way to see all the monuments lit up.

In NY, I did a food tour at the Chelsea Market that I loved - if you love to sample different foods, I recommend this company!

I would recommend the 7-day Metro card for NYC. If you do 12 trips (which in your case will be 2 or three trips a day) you will save money. I agree that you are too heavy on the free tours, but that is a personal preference and incredibly enough yours may be different, but they are not 'free' as you pay what you want. It would take a heart of stone cold coal to pay nothing.

A bit of research will show you where totally free park ranger tours are happening, for example at The Vietnam War memorial, in DC.

I'm not sure when you are visiting but if you are able to book the Crown of the Statue of Liberty I would recommend it. The tenement museum recommended previously is awesome as is the African American culture and history museum in DC but bookings for the latter, while free, are much sought after.

if you are able to book the Crown of the Statue of Liberty I would recommend it

Just be aware that these sell out months in advance.

the African American culture and history museum in DC but bookings for the latter, while free, are much sought after

There are "same day" tickets, but lines can form by 6 am, and your timed ticket may not let you in until 4 pm (assuming you get one; they sell out fast). Hopefully the crowds are lessening a bit by the time you get here, but this is still a very new and popular place.