From my experience of healing when the tank jumps over to the other ship, I have found that DKs are the hardest to heal there.
Typically we will two heal this fight, but when we started we had three healers and found that in order to keep up a DK we had to have all three of us spam healing the DK. Now we are all a lot better geared and with a decent holy pally the other healers don't need to worry about the DK. The main difference in terms of gear for us is that now all our holy pallies (and everyone else) are sitting between 5.3k and 5.6k or more GS which is a big step up for healing output from 4.6 k.
With the mana and SP that you currently have you shouldn't be expected to solo heal the tank that jumps across and other healers should be helping you, whether by shielding or by direct healing. In the case that you described, the disc priest should have been on the ball and seen that you needed help before the wipe.
As to whether you could handle raid healing alone, very simple answer: NO.
There is a reason for having tank healers and raid healers; tank healers heal the tanks, and as Lecan says '... our extra tools are all very good for keeping two/three people alive through very heavy damage for a long time ...', while raid healers have abilities to heal large numbers of people quickly who have a lot smaller health pools and who tend to take less damage. But tank healers should also be helping out with healing anyone else when time permits, and raid healers should be treating the tanks as part of the raid and healing them as needed.
There is also the issue that in some fights healers can be stunned / silenced / etc. and the healers need to be available and aware enough to pick up the slack when this happens.
On to whether better gear making healing easier, again a very simple answer: YES.
Simply put better gear means more mana, more SP, higher crit chance and faster casting (more haste). As I said earlier we used to three heal in ICC, but as we've geared up we now two heal.
And for your final question:
Lecan wrote:
To answer your last question, we are not as good at raid healing because we have basically no AoE heal and Flash of Light isn't enough to keep up with most of the damage in progression raiding. We are, however, very good at tank healing: Beacon of Light, Holy Light, mana longevity and our extra tools are all very good for keeping two/three people alive through very heavy damage for a long time.
Whether you keep playing your pally, or drop it and play disc is something no-one else can answer, it depends totally on what you enjoy and your playstyle. However there is nothing wrong with trying a disc priest as well (or any other healer) and seeing which you prefer, or like me have three ICC healers and choose which one to play to complement the rest of the group.
TL;DR; I gave great pearls of wisdom and you missed them :(
(I admit that the pearls of wisdom may have been in other posts, or more likely after closing time at the pub)