Valencia - where to stay, what to do

You can easily see Valencia in 3-4 days so unless you´re planning to do daytrips outside the city or hang around the beach a lot (which will be overcrowded in the summer) I´d say it´s too many days in one place.. Location in the city depends on what you´re interested in. The downtown is called el Carmen, another lively neighbourhood is called Ruzafa/Russafa; if you prefer to be close to the beach you´ll find a lot of wide range accommodation options. In any case, the metro system is quite good in Valencia and it´s not a huge city so you´ll be able to get around and from one place to another without too much hassle independently of where you´re staying, except maybe around City of science and arts (Calatrava buildings), which is fairly badly communicated to the rest of Valencia. Generally speaking, I would look anywhere between the center and the beach.
As for what to see and do, it again depends on your interests. The typical highlights are the old town and surroundings (the cathedral, the central market, la Lonja, torres de Serrano, plaza de la Virgen, plaza del Ayuntamiento, the train station..); Turia, which is the old river turned into a park and is a good green area that runs throughout the city; on one end it has the Oceanographic and City of arts and sciences, on the other end Biopark (the zoo). I´m sure you can find all this and more on google. Best places to have a paella are either next to the beach (can get very expensive) or outside the city – El Palmar, Pinedo..

Plan was to visit Valencia, Cartagena and Granada but then I realized there is no good connection between those cities.
As I am traveling with a small child, everything should be easy and without any additional stress.

My interests are history, museums, old city, markets and everything worth seeing. Parties and night life is something I cannot experience now and I don't have any need for that with a child.
I would like to find accommodation somewhere that is near the center so we can go for a coffee or breakfast and get by foot to see downtown or a museum. If you can give me some names of district so I can search on web it would be great!
You suggested Carmen and Russafa, shall I search there?
Of course we would like to go on a beach but its not a priority, and I didn't mentioned... we will got there during the August.

We took our daughter to Valencia for her first birthday and had a great trip. The Bioparc was a big hit - I am not a big zoo person but really enjoyed it. The Turia park was a great place for her to walk/crawl around. I don't know how old your child is (ours was too young), but there is an amazing playground in this park.
We didn't make it to the City of Science because as alluded to it's awkward to get to by public transport. We did go to the beach, but it was November and mostly deserted. I wouldn't call it the most attractive beach I've ever seen, but there are many many restaurants/cafes along the stretch.
Russafa is a slightly grungy trendy/hipster area which would be of little interest to a child. We stayed in El Carmen and liked it a lot.

yes, unfortunately Valencia and Granada are not very well connected, if you´re not renting a car it would mean very long and exhausting bus rides.. I guess your best bet is Carmen, you´ll have to reserve accommodation well in advance.

What is the situation with Cordoba? Is it possible to get there without any difficulties?

Three/four days will indeed be enough.
If a good walk is no problem, just go to 'Calatrava' by taking the rio Túria. A gem by itself.
Alternatives for the extra destinations: Alicante and Elche.

badly communicated amscams ? Have you been speaking too much spanish lately ? :-)

haha probably

If I´m not mistaken some of the fast trains (AVE) that go from Valencia to Sevilla stop at Cordoba, it might be an option for you, you can look at the timetables and buy tickets on renfe.es (you have to have a lot of patience with that website though, it´s very slow and not very user friendly).