Well, the question of "What is Product Marketing" Could mean different things at different companies, but my answer is that we provide the voice of the market and the voice of the customer internally to the product manager so we can build products that resonate with our audience, and we are the voice of the product externally providing the appropriate messaging and positioning to go to market.
Congrats on the new role! Very excited for you. I agree that it is good to have a 30-60-90 day plan and to make sure you can show progress and positive impact early yo make a good impression. That said, I would suggest you give yourself some time during the first 30 days to absorb as much as you can about the company, the interpersonal dynamics, the challenges and opportunities so you can then define and priorities in month 2 and deliver something of value in month 3 on the top 3 opportunities you identified in month 1 and worked on month 2 and 3.
Make sure you have periodic meetings scheduled with each separate and some times with both together. As PMM, we bring the "voice of the customer" and the "Voice of the market" from the outside in, and we provide the messaging and positioning for the go-to-market strategy for the "inside out", so we need to get sales input and feedback intro product to influence the roadmap, and make sure we educate sales on the product in the best way possible so it resonates with their customers.
One of the main skills I see to success in PMM im Empathy. Empathy in the sense of being able to to put yourself in other people's shoes. You are the customer and market advocate internallt and the product advocate externally, so understanding those different perspectives can help a LOT in any PMM materials you are developing, from slides to demos to websites to campaigns.
This question reminds me of Reid Hoffman's book "The Startup of You". If you think of yourself as a startup, think about which skills you have and are good at, what others you want to learn, and see how you can hone on those skills in your current or upcoming role as well as which other skills you want to learn or develop so it will help you get the next role. There are different career paths for product marketers, so you have options.
I am more about roles than hierarchies, and Sr. Manager or Director may mean different things at different companies depending on their size and level of development / maturity of the company. I would be more focused on the role, increasing responsibilities and impact than titles so you can write your own story in terms of career progression. That said, in any company you will need to be already performing at the next level to make the jump. So I would look at the Director of Product Marketing job descriptions at companies of similar size and development to the one that you are in, and make sure you are performing at that level to then go for that promotion.
It makes sense that the first marketing hire on a product-led organization is the Product Marketing lead :). I would say put your strategy together and depending on the priorities you will see what kind of marketing hires you need, vs perhaps going with external agencies for some marketing activities. You may want to keep the strategic items in-house like competitive intelligence, but anything related to marketing campaigns you could start with external agencies / contractors with the optionf or them to become full-time. It really depends on your audience and what you are trying to do next. For example, if you are targeting developers, my first hire there was a technical writer to clean up the product / API / SDK documentation. So sorry to say that the answer here is "it depends", but hopefully the context above can provide some pointers.
Product Marketing could be under Marketing or Product depending on the organization. The good thing about a Product Markering role is that the role, by its nature, interacts with many different parts of the business, from Sales to Marketing to Product, so Product Marketing gives you an opportunity to see how the organization works beyond product marketing and prepares you well for leadership roles accross the organization.
The definition of "Extensive competitive product research" may be different for different people. I suggest asking the CEO and Product / Engineering teams the kind of questions they are looking to answer. Sometimes the high level market research you can get from a 3rd party will not be enough, and you will need to get creative to get the information needed via surveys, primary research or other methods. My best advise here is to define the task in more detail to undertand what people are expecting.
Product Marketing is an evolving role and can mean slightly different things at different companies, from sales enablement to competitive analysis and everything in between. Hence, I would suggest this product manager to define what product marketing means to them and find / build a place most aligned with what they want to do.