Logistic regression vs GLMM

#1
Hello,
I'd like to ask you for help with a choice of correct model for my analysis. Here is the problem:
1.) I should find out whether the risk of stroke after 65 age depends on gender
2.) Find out whether the association of gender with stroke depends on age
3.) Can gender effect on stroke be explained by differences in known risk factors( such as high blood pressure, smoking etc. ) between men and women?

Data consists of 3954 rows(3954 patients) and 13 columns(13 variables).
In year 1990 3954 patients were recruited and were followed until 2002. New cases of stroke that occured during this period were recorded and verified.

(Important)
Variable incstr contains information about the occurance of stroke during the study. Code 0 means the subject didn't have stroke either before or during the study.
Code 1 means the subject had stroke during the study.
(Code 9 means subject had stroke before the study-these observations are unimportant for our study-delete)
The variable agest is age in which patient entered the study.
The variable agex is age in which patient left the study. The exit was determined by the earliest of the following events: the first stroke, death, refusal to continue.
agex-agest is the total time spent in the study.

What model should I use to approach this problem? Here is my thinking.
1.) I will identify risk of stroke as a chance that a man will have a stroke within the next 10 years(that was the duration of the study).
I will delete observations for the patients that died any other way, or left the study without having a stroke.
Then I will use logistic regression for response variable incstr and as the regressors I will put all the other variables(except agex).

2.) Make the data as yearly observations (each year observing a patient had or hadn't stroke that year) and identify chance of stroke as a chance that a patient will have a stroke in the next year?
This has a problem with independance I think. Y_ij:=[i-th patient had a stroke in j-th year of the study]. If we take P(Y_ij | Y_ik=1)=0. So I can't use logistic regression for it.

Please post your thoughts about what model I should use. Best with explanation.
Thank you very much in advance. :)