Answers
1) Strong Correlation/association but not significant
YES, it is possible
2) Significant but NOT Strong Correlation/association
YES, it is possible
3) Examples
1) generate X and Y variables with a correlation of 0.50 and sample size of 7
run a pairwise correlation and get p-value
* result r=0.5570, p=0.1940 - Significant
AND
2) Rerun this whole thing exactly the same way but use sample size of 100
* result r=0.50, p< 0.0001 - Not Significant
Basically it depends in the sample size. Sometimes you may have corr = 0.001 and sample size is say 100000 then in that case it is weak correlation but it will be significant due to sample size.
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