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So I know that many people go to DC to take pictures as well as visit museums.

Unlike others, I plan to be up early and catch the sunrise for the National Mall and be around that area early (5-6AM) and later in the evening (9-10PM or later) for the night. I know that terrorism has created a lot of things like instilling fear into people.

Are there any safety problems (e.g. getting robbed or security guards telling me I can't be taking pictures during those times) that I should be aware of? I ask this since I got asked by a few people when I was part of staff taking pictures during a fairgrounds for their advertisement campaign. I haven't been asked in San Francisco so I assume it would be the same but, last I remembered, isn't Washington DC is a special district? I'm not sure if not only the usual street smarts but also rules apply or maybe they don't and people with large backpacks are deemed dangerous?

Update1: I plan to be there during the winter if that makes any difference in the next few weeks or so. I will have a large backpack and also tripod if that helps. I will be taking pictures both during the weekday and weekend and be in and out of the area.

The National Mall is officially open 24 hours a day, with rangers on duty from 9:30 a.m. to 10:00 p.m. The rangers are there to support visitors by answering questions and interpreting the landmarks, and are not primarily responsible for security.

That falls to the U.S. Park Police, with neighboring areas patrolled by the Secret Service Uniformed Division, the Capitol Police, the Federal Protective Service, Smithsonian security (yes, there is such a thing), and many others, plus the local law enforcement agency, the D.C. Metropolitan Police.

There is almost no violent crime around the National Mall… almost. From time to time there is mugging or assault, which then makes headlines (it does not when such crimes are committed elsewhere in the city). As with anywhere in the world, at any time of the day, at any time of the year, however, ordinary prudence should always be heeded: be aware of your surroundings, stick to open and well-lit areas, avoid being isolated, be mindful of valuables.

As a local resident, I will say that in the summertime, the hours you cite would be entirely unexceptional; you will find joggers on the Mall starting at sunrise, and tourists lingering past midnight at the memorials. In the winter, naturally, crowds are much thinner overall and especially after dark.

As for photography, the National Park Service only requires a permit for commercial photography or filming (43 CFR 5.2):

(a) All commercial filming requires a permit.

(b) Still photography does not require a permit unless:

(1) It uses a model, set, or prop as defined in § 5.12; or

(2) The agency determines a permit is necessary because:

(i) It takes place at a location where or when members of the public are not allowed; or

(ii) The agency would incur costs for providing on-site management and oversight to protect agency resources or minimize visitor use conflicts.
(c) Visitors do not require a permit for filming or still photography activities unless the filming is commercial filming as defined in § 5.12 or the still photography activity involves one of the criteria listed in § 5.2 (b).

If you are in fact a commercial photographer (you are either being paid to take these photos, or you intend to sell or license them), you would do well to read the entire rule, including replies to public comments. For example, some of the statues in the war memorials are copyrighted works, and so photographs cannot be commercially distributed without permission. Otherwise, any restrictions (for example, on the use of tripods) will be indicated with signs in and around the particular facility.