I grow climbing nasturtiums every year. I love them and so do bees. As others have said they attract butterflies that lay eggs and lots of blackfly, keeping them away from my other plants. So that they don't get out of control I take off the leaves that are heavily infested with eggs and put them in an isolated spot in the garden with other leaves I've pruned to allow them to grow.
The blackfly larvae I remove by hand or with a water jet, or if particularly heavily infested, often on stems close to the ground, I remove the stem and put them in the bin.
I also take off a lot of leaves to reveal the flowers, it doesn't harm this prolific plant at all. But it does need regular deadheading to encourage new flower growth. You've already seen the corms that are the seeds. It's better to remove the flowers before they get to this stage, when they curl over and often look a little ragged.
It's not possible to deadhead enough to stop seed from being produced so just remove them when you see them. Deadheading and bug removal can be very time-consuming if you have a lot of them as I did last year when I had a fence and a wall covered with them. But the display was worth the effort. Although, one drawback as someone else mentioned is that they self-seed very easily and if like me you grow a lot, you end up pulling out lots of nasturtium shoots throughout the year.
If you grow them on a fence, trellis, or near bushes, you can weave the plant through them to create a magnificent display. If you do that be careful because the stems are very easily snapped because they are full of water and fragile. I just let them grow as long as possible, which makes them easier to handle.
The final thing to mention is that the flowers and leaves are edible and are lovely in salads. The large leaves get quite bitter so choose the smaller leaves. The corms, or seeds can be pickled and are similar to capers.
Oh, and regarding your question as to how to keep the seeds. First let some of the corms grow as large as possible, which is a little smaller than the first joint of a thumb. Then cut them off where they join the stem, store them in a a paper bag or envelope in a cool place until you're ready to plant them. I often put mine in the salad box of the fridge, or in the loft. But any cool place will do.