How easy is it to get additional training and jobs in active duty?

Get “your 68W”? It’s an MOS not a school. You can become a 68W in the NG or USAR by switching MOS.

There’s not such thing as being a jack of all trades on the medical side. You are qualified in what you qualify in. Those qualifications are civilian qualifications so you’ll need to put in work for those. The Army will send you to schools that will qualify you for a job, but not all jobs. You need to identify the medical route you want to take first. Doctor, nurse, PA, medic/Paramedic , LPN, Physical Therapist, Nurse Practitioner, etc are all different jobs with different routes. You can’t be all and one at the same time. 68W is a great job for learning the other routes, but you won’t receive much advanced training from it.

Not trying to crush your dreams but let me break it down. 68W is a MOS. You can get that MOS provided you qualify for it, the slot is open, etc. Even then there's a lot of schools you can get as a 68W but not necessarily meaning you will. Flight medic for example. Not every 68W can go to that course. I have a good friend of mine who's a flight medic in Korea I can tap him to provide answers for questions pertaining to that. There are also other courses depending on the set up of that unit. For example, an airborne infantryman in 82nd isn't likey to get sent to a Bradley operator course because Airborne units don't operate Bradley's. Same soldier can PCS to Bliss or Hood and immediately get assigned to a unit that does.

You won't be a jack of all trades. Infantrymen kinda comes close in a certain respect but even then it's more of a knowledge in multiple aspects, not mastery level. Also keep in mind school seats are based partly on funding. You aren't the only hero that wants schools. There's OMLs on who attends schools.

Last piece, is yes. I feel you're chances of getting schools in general are better on active duty. That being said, you won't get every you ask for.

This is one of those questions that are extremely difficult to answer because of the variables. "Seats" at the schools are funded based upon anticipated requirements in the budget process. Filling those seats reflects the services needs during the actual budget year. Sometimes the budgeted number of seats is in excess of actual requirements and the schools and monitors scramble to fill quotas; other years the need exceeds budgeted numbers. The tempo of operations of your unit also effects your opportunities. If your unit has a high tempo of operations, they will be hesitant to even request a seat for you because you are needed to work in your specialty.
The need to meet their mission is the priority of your unit. Allowing you to train outside of your specialty is a hit or miss situation. If someone in your unit believes in such training and is willing to work through the bureaucracy, it is possible to pick up available seats, but the stars must align correctly and it requires effort on your part, and others, both within your unit and outside of it to make it happen. Don't make any career decisions based upon "working the system" to achieve your desires.