We are both citizens of a non-EU country. Her new employer took care of German employment authority approvals and she will apply for a national visa next week. My current company is looking for a job for me in Germany, but it will take some time. I am also considering a career change so I am applying for some jobs myself as well.
The questions are:
Should I apply with her, basing my application on her work visa?
Would it be a problem if I get the German national visa while continuing to work for my current employer in a non-EU country for at least a couple more months?
You can apply together with your wife, why not. Usually (but not always!), it is suggested to get the same job permit conditions as your wife has. E.g., if she is allowed to work as an employee, you will be allowed as well with a high probability (however, you won't be able to create your own company).
Still, there could be some problems, please see this page for details. In the end, it depends on your personal case, and I suggest you to double-check it. But, do not be scared, as I wrote, usually, you are expected to get the working permit if your wife got the residence permit for the goal of employment.
I'm not sure what you mean here. You can get your national German visa valid from "2 months later" and continue working for these 2 months in your current country. You can also get a visa valid from "now" and stay these 2 months in your current country (the visa should be valid for 90 days), but then you will have a limited time (the remaining 30 days) to move to Germany and apply for a residence permit.
However, even in the latter case, you can stay in Germany after your visa has expired (if you applied for the residence permit before the visa expired), until the decision about your residence permit will be made. You can also try to apply for a visa valid from "now" until "6 months later", but only for 90 days within these 6 months. Finally, if you want to work for your old employer remotely from Germany, then I'm not sure how it is regulated (this also depends on the form of your employment, and things become more complicated with taxes etc.).