There are so many things to write about on this, let's start with:
The Doctor encounters many different alien races and there is no reason to believe that there are not intelligent civilisations out there. Indeed, the Doctor is an alien. I don't believe that the majority of aliens, should they exist, would look like humans but in the Doctor Who world it makes life a lot easier. I think the best aliens are the ones that are very different to humans, such as the Daleks (my personal favourite). It's great that they dealt with the language issue by having the TARDIS translate everything.
Sonic just refers to sound and there are already many devices that use sound, so in theory a sonic screwdriver doesn't sound too bad (excuse the pun). However, the Doctors sonic screwdriver does much more than unscrewing things, it can control the TARDIS, it can scan instruments, unlock doors. In fact, a whole host of stuff. Could we have something like that? No, not very long time, if at all.
The Time War
If a species were so advanced they could time travel, then yes a time war could be possible where you try to change history for your enemy. In Doctor Who it was the timelords that started the time war with the Daleks by trying to destroy them at the point of their creation. It's an interesting concept, but it does assume that even a very advanced civilisation has mastered time travel.
The most wonderful spaceship ever seen in science fiction. The ideas behind the TARDIS are so far into the future, if at all, that it seems like magic. To be bigger on the inside than outside (by having the inside in a different dimension to the outside of the ship), to be powered by the "Eye of Harmony" using solar mechanics (engineering on the scale of stars), to be able to change shape to blend into the background (although the Doctor's ship has not changed shape in a long time) and finally to have a mind. It's this last item that there has been progress on in real life, we are developing artificial intelligence - although it is at a very basic level.
Who knows what Governments around the world are doing. I do know that until recently the UK Ministry of Defence had a person employed to review reports of UFOs. This has now stopped and the reports have been released into the public domain. I don't know what other countries do, but I doubt very much that Area 51 has aliens in it, or alien technology. I could be wrong!
One of the main reasons Doctor Who has survived so long was that the Doctor could regenerate into a new form (new actor) and live on forever. In reality everything must die, even though some lizards do have the ability to re-grow their tail etc. However, the human body constantly regenerates over time. If you're about 20 years old then there is very little of you that still survives from when you were 4 years old. That regeneration process becomes less efficient as you approach old age (which is why there is an old age in the first place). With genetics, that progressive lack of efficiency could be slowed so that the onset of old age comes much later. I don't think it will be that many decades before mankind has the technology to increase lifespan to more than 200 years. However, perhaps the technology could one day be there to make the human lifespan considerably longer.