1. I'm not an EE, but that has been (in my opinion and that of others I've read) the most steadily in-demand technical background for patent agents and attorneys.
I've never heard that it was difficult for a BSEE vs. an MSEE. Of course, as between two otherwise identical candidates for some given position, the MS might be looked on more favorably, especially if they concentrated during the MS work on something of interest to the prospective employer. But on the whole, BS should be fine.
2. I don't know much about expected EE-as-EE payscales. I do know patent agents with some experience who are making 120 or so. I also know some in smaller markets who are making more like 80. (None of the ones I know are EEs, by the way.) Wait and see if others chime in.
3. It would be at least a bad idea if you were a BS Biology or Chemistry, as those don't seem to be in demand. It's not a bad idea for an EE.
Another thing you can do is start floating a resume now, while studying for the patent bar, to get a feel for the market. Firms do employ pre-barred engineers (usually referring to them as "technical specialists", and sometimes as "patent scientists"). They do much the same work as patent agents (application drafting, office responses) but are ghosting the work for someone who is a registered agent or attorney to review and file. If you find strong interest, you could make the jump earlier than planned if you like what you hear about their expectations for your initial work, by when they expect you to get the exam done, and what will happen once you do. (Pay bump? Or would they start you at a salary similar to an agent? Maybe others can comment on this also.)
Something else to consider is looking at who works in your company's patent department. Do they employ any agents, or only patent attorneys? (Varies pretty widely by company and "culture" in a given company's law department.) If they do use agents, would they consider taking someone from R&D into the legal department? Of course, you'll have to be the judge of whether just making soft inquiries would upset the apple cart with your current management.