This is a great question about both politics and history. It relates directly to the changing nature of what a "country" is. First, let me directly answer your questions:
- Given that France, Germany and the UK have larger economies than that of Russia, why do they spend less in defense than that of Russia and always seem to be scared of Russia?
There are two easy answers for this, namely a) Because the US military is present in the region, so the people feel secure from Russian attack and b) because the voters in these countries vote for social services like health care instead of military spending.
- Why does Russia maintain so massive military when these countries can't?
Germany, UK, France, and other western European countries are more than capable of spending a ton of money on military. They choose not to because they are working as a team within the Nato alliance - and they do this for a couple of reasons that will be elaborated below.
- Why can Russia spend so much on defense?
Because Russia can do as it pleases within its means!
As for why it does, the reason is the same as for other countries: they spend the amount they can afford for its constituent population to feel "secure". Russia is a huge country with strong neighbors. In the past 100 years, they have fought wars along most of these borders. While the western European invasion during WWII was significant, it is notable that there was a small hot war with China in the '60s and '70s, and war with Afghanistan. They still have territorial disputes with Japan - and they are slowly being intimidated by an expanding Nato that is vocally hostile to the Russian state and hence the Russian population.
- Given that these countries spend so much on defense, Why do these countries seek US help/alliance to counter Russia?
This is the most critical question you ask, I will answer with a counter question: do these countries have a choice?
At the end of WWII, Europe was largely occupied in the west by the US military, and in the east by the Soviet military. One could say that all the countries in the East became client states of the Soviet Union, and that those in the West became client states of the USA. Both the Soviet Union and especially the USA, maintained large military bases in these states. The US still has numerous bases. Could we call the US an occupying power? What would it take for Germany to end the presence of the US military on its soil?
However, most Americans and Western Europeans bristle at the idea that they are involuntarily "occupied" by Americans, they are instead "Allies"! And it is good, of course to be best friends with the rich American business interests. Trade and mutual economic benefits keep Germany, France, and the UK, most of Europe, in this alliance. Indeed, besides a handful of nations like Iran, North Korea, Syria, Cuba, Venezuela, and Bolivia, everyone tries to stay on the good side of the Americans. Both for commercial reasons, and because the US, frankly, has a dominant military presence just about everywhere. Perhaps a commenter can correct me, but I think for some reason the US has ~5000 military bases overseas.
- Why is NATO still so important to them even though they spend so much on defense? Aside the USA, these four countries' combined spending far exceeds Russia and Russia still can bargain with them in Syria.
NATO is important because it indicates what relationship these countries have with Washington. Russia is in Syria for a variety of reasons; it has more interest in the area than Syria's former colonial masters, the UK and France.
An important point to note is that the concept of a country is fairly new; I have read, and agree, that modern states did not become such an important unit of human organization until the Great Depression in the 1930s; at that time the problems facing them - broken economies and socialist movements - the country level government was the only organization existing that could solve these problems while the existing power structure stayed in power. Since then, states are portrayed as "immortal" and "ever lasting". But in truth, they are not. Competing levels of organization - family, village, community, province, supranational, class, or something else - can eventually take primacy in terms of absolute power. What will happen is a matter of conjecture, but the point is clear: countries are not the only level of organization.
For example, what is the EU? Are Germany and France and UK independent countries, as the UN says, or are they members of a single supranational organization? Both really, for example, the UK will have to negotiate its secession or pay a heavy price. While everyone can agree that in name these countries are independent, they do heavily coordinate most of their economical and political activity. This includes military activity.
The level of military coordination between western European countries and the US means that spending on military need only match Russian military spending on a collective basis. I think with data about US spending and presence in and around Europe, the collective manpower and economic investment will be far greater than that of Russia.