1. Should the JR pass be activated on the day to Mt Fuji?You could take the JR Chou Line from Shinjuku to Otsuki. That would be covered by the JR Pass. However you'd then have transfer onto the Fujikyu Railway Line to Kawaguchiko Station, this isn't covered by the JR Pass.
Tbh I'd recommend taking a highway bus to Fuji Five Lakes instead, that is what I would do.
You can find more information here though
http://www.japan-guide.com/e/e6905.html2. Is the JR pass the better choice to head to Mt Fuji, or should I get the Fuji-Hakone pass instead, and combine both visits.If you are dead set on visiting both Hakone and the Fuji Five Lakes then I'd recommend doing both during your second stint in Tokyo between 15th - 20th May. Because the Fuji Hakone Pass will give you 3 days of consecutive travels between this areas and Shinjuku.
3. We are interested in Lake Yamanaka and Lake Kawaguchi, but based on our travel plans, do I have to choose one or can I do both? You might be able to at a push, but you'd most likely end up rushing your way through that day, leaving yourselves unable to enjoy what is on offer. I'd recommend doing one or the other. And just wondering if you have heard of Lake Saiko?
4. Can I fit Takayama into my itinerary, or will I be stretching it thin?You would be stretching it very thin if you tried to include this. From Kyoto to Takayama the entire trip would take around 3 hours (one way). That's 6 hours in total that you will be spending just sitting on a train.
Honestly, I'd recommend just enjoying the days that you have in Kyoto.
Thanks, for giving me information though on your varied ages.
In Osaka there is a large number of places that I'd recommend. Dotonbori, which is a amazing area for Osaka street food such as Takoyaki & Okonomiyaki and the atmosphere is great. The best time to go (imo) is during the evening.
There's also places like Osaka Aquarium and if you are into it, the Harry Potter World at Universal Osaka.
Whilst as a day trip from Osaka I'd recommend visiting Himeji Castle in Himeji and then Kobe on the way back. In Kobe you have the Shin-Kobe ropeway that offers wonderful views of Kobe and Kobe Harborland offers a varied mix of shopping and restaurants.
I'm guessing you have already looked at places to visit in Kyoto? Or would you like some ideas?
For shopping, especially in Tokyo there are a large number of different and interesting places. Harajuku, Shibuya, Nakano Broadway, Akihabara and Odaiba are all fantastic places to visit.
I would also recommend visiting Yokohama.