(I'm a Director-level with 30 reports in my organization, with a few managers as direct reports. These thoughts below reflect my current problems as well as problems I had when I was "just" an engineering manager. Note that the problems don't go away, just increase in scale...)
Number one for first two questions: hiring. Finding solid, dependable people with the right skills and attitude is really, really hard. I probably spend 30-40% of my time on the hiring side of things. We have so much work to do and not enough people to do it, and combining that with the slow rate of inbound high quality candidates means that I have to spend a ton of my time screening and talking to candidates. Someone else mentioned turnover, but I have never (so far in 6 years) had anyone quit while working under me, so turnover for me has been really low.
Number two problem relating to the above: diversity. It's nigh impossible to find women and other marginalized groups. They're in such high demand and the supply is so low that it's just so hard to hire people and have your team not look like a team of white dudes.
For the third question, I talk to my old bosses and coworkers the most. I have a fantastic relationship with my last two bosses. Nowadays we're peers (same title, different companies) and we compare notes and mentor each other. If you don't have someone like this already, I suggest going to meetups (post-COVID) and meet other engineering managers. A shortcut to this is to find a new job, and then your old job colleagues can be your external mentors ;) That said, there's nothing wrong with having mentors within your company. Just be up-front with them about what you're looking for, especially if they're upper management.
For the fourth question, I've never really found that tools have an impact in either direction. I've yet to find a tool outside of Excel/Google Sheets and email that is indispensable.
Number one for first two questions: hiring. Finding solid, dependable people with the right skills and attitude is really, really hard. I probably spend 30-40% of my time on the hiring side of things. We have so much work to do and not enough people to do it, and combining that with the slow rate of inbound high quality candidates means that I have to spend a ton of my time screening and talking to candidates. Someone else mentioned turnover, but I have never (so far in 6 years) had anyone quit while working under me, so turnover for me has been really low.
Number two problem relating to the above: diversity. It's nigh impossible to find women and other marginalized groups. They're in such high demand and the supply is so low that it's just so hard to hire people and have your team not look like a team of white dudes.
For the third question, I talk to my old bosses and coworkers the most. I have a fantastic relationship with my last two bosses. Nowadays we're peers (same title, different companies) and we compare notes and mentor each other. If you don't have someone like this already, I suggest going to meetups (post-COVID) and meet other engineering managers. A shortcut to this is to find a new job, and then your old job colleagues can be your external mentors ;) That said, there's nothing wrong with having mentors within your company. Just be up-front with them about what you're looking for, especially if they're upper management.
For the fourth question, I've never really found that tools have an impact in either direction. I've yet to find a tool outside of Excel/Google Sheets and email that is indispensable.