When variables are co-linear, you can think of them sometimes as being different manifestations of the same thing. Say I had a dataset of happiness of cats, a variable of whether they were soaking wet, and a variable of whether or not there were nearby children who thought it was fun to throw cats into water. Clearly cats don't like water, yet sometimes they will fall into it on their own. More often however, they are thrown in by malevolent children. Sometimes however, malevolent children fail to thrown cats in the water.
So, wet cats and malevolent children are different, but can be thought of as a unitary dynamic. If a researcher was only interested in the effect of wetness on cat happiness, and didn't control for malevolent children, the estimates would be biased. Include them, and VIF goes up. This is because you simply don't have enough independent observations of wetness to know its effect apart from the effect of malevolent children.
Shrinkage estimators are one way to go. Basically, you increase the bias of your estimator in order to decrease its variance. Appealing for prediction, but not for inference.
If you're willing to put aside (or think differently about) inference on individual model terms, you could first do a principal components analysis, "interpret" your principal components somehow, and then fit your regression to the rotated dataset. Collinearity will be gone, but you're only able to conduct inference on these PC's, which may or may not have a convenient interpretation. In the case of wet cats and malevolent children, the first PC would increase as the probability of wetness got higher and as the probability of malevolent children increased. The other PC would be perpendicular, and relate to wetness as the probability of malevolent children decreased. If you simply wanted to know the effect of wetness absent malevolent children, you'd be interested in the coefficient on the second PC. Most PC regs don't have interpretation this straightforward however.
It is also worth emphasizing that prediction from a model with high collinearity is fine. So if your F-stat is good and you don't care about any of the coefficients individually, leave the model as it is.
The fact that additional variables make large changes in your model doesn't mean there is collinearity nor does it mean it isn't "robust" (although I guess that depends on what you mean by "robust"). Nor does a reasonable VIF and a high F mean that additional variables won't have an effect. Only completely independent variables will have no effect on the other coefficients, and variables can be a long way from independent without having high VIF.
It means that controlling for those additional variables makes a large difference in the other relationships.
You haven't said what any of these variables are, so it's hard to be specific. However, let's imagine simpler models:
Model 1: log(income) as effect of race/ethnic group
Model 2: log(income) as effect of race/ethnic group + age
The coefficients in both models will likely all be significant (if it's a reasonable sample size). But the coefficient of race/ethnicity will (I bet) drop in the 2nd model because the average age of different ethnic groups is different, and income tends to be related to age.
EDIT in response to edit in post:
Since you have country level and company level variables, you should not use OLS regression as the data are not independent. You need to account for this. One way is with mixed models (aka multilevel models). I don't know how SPSS does this, but in SAS it would be PROC MIXED and in R it would be nlme or lme4.
Best Answer
As I understand it, you have a variable that is the average revenue of sampled companies in different countries. The problem with having the different countries using different currencies is that they are no longer comparable at all.
The regression model has no knowledge that these cases have been measured on different scales and so would, as a basic example, treat an average revenue of 200,000 Yuan (which is actually only around USD$29, 000) as larger than USD100, 000.
Thus, you will want to convert all of your values into a comparable value such as converting all of them into USD or maybe converting all of them into some measure of spending power (I'm not an economist so I don't know what metrics are used for this but maybe something like how many loaves of bread the revenue would allow you to buy).
Edit: I am not familiar with the concept of lagged assets so I'm not sure whether what you refer to there is converting the company revenues into some comparable metric.