In terms of dog/cat-friendly apartments in the center of DC, look at these buildings (if you have a cat, it's probably a little easier to find a place, but I wasn't sure...):

-The Shawmut Apartments (at 19th + Columbia, on the Western side of Adams Morgan). This is a nice vintage building on the quiet side of Adams Morgan, which is very convenient to Dupont, as well. It has very good transit connections, groceries and a gym nearby, and a nice park across the street for walking your dog (if that's what you have). A studio there would probably be somewhere around $1000/month (not sure...)

-The High Towers (1530 16th St. NW, near 16th + Q, in Dupont). This is a very nice older building, as well. Studios here might be a bit pricier (I'm really not sure what they go for...), but it's a very good/convenient location with good transit access, and shopping/groceries nearby.

-The Barclay and Ravenel (1610 + 1616 16th St. NW, between Q + Corcoran Sts, in Dupont). These are two older buildings that are pretty basic (and might be more roach-prone), but very convenient (great transit access) and cheap, and at least one of them allows dogs. Studios would probably be in the $900-950/month range. Shopping/groceries/gyms nearby, too.

I doubt that any of the above have parking (in fact, I know that the High Towers, Barclay and Ravenel don't), but if you can ditch the car (easy to do without if you're also working downtown or near Metro), they're decent places. I live next to the High Towers, and a block from the Barclay/Ravenel, and it's a great neighborhood.

Otherwise, the suggestions above about basement apartments in Capitol Hill or searching on Craigslist are good ones. Also, look at the apartment listings in the Washington City Paper (washcp.com).

Welcome to DC, and good luck with everything.