1. It seems that springs are too hard and sometimes when i push the router down it stays in that position. Should I put some machine oil and lubricate the springs and axis.
That depends - if there are oil-lite bearings or collars (they are brass colored bearings), they can be cleaned but should not be lubricated. Machine oil will (or any lubricant) will hasten the failure of the bearings or collars. These are manufactured with lubricant impregnated into the material and only require non chemical cleaning (dry cloth). The shafts and columns can also be dry cleaned to avoid rusting. The sticking is probably due to dust collection.
It is unlikely that for the life of the router these bearings would fail, lubrication would only help accumulate more dust and more frequent cleaning of the bearings or collars would be experienced.
Hope this helps.