1. Based on your skills, you can do the following for money. Though not the best, but decent, especially at your current levels. You can chop oak logs near Varrock west bank, You can kill cows for cowhide, north of Taverly bank. For additional profit, you can convert all the cowhides into hard leather at Al Kharid. You can mine iron ore at Al Kharid or Falador and bank them. You can mine or buy clay from the G.E and using the fountain at G.E to convert the clay into soft clay and selling them.
2. Your next upgrade would be Rune armor, which is the best armor for F2P. It requires 50 Defence to equip. You can buy the entire set at the G.E. for about 140k. If you killed cows from 40 Defence till 50, you should be able to get enough money from selling the cowhides to buy rune armor.
3. Sadly, no.
4. There are certain skills that can be bought such as Cooking, Smithing, Crafting, Prayer, etc. Of course, the more you spend on the skill, the faster you level it up. If not, there are slower ways which are cheaper, if not free. Certain skills cannot be bought but rather trained by grinding such as any combat skill, Fishing, Woodcutting, etc. To train fast, you'll have to keep going at the skill.
5. There are a lot of things that you can keep in the tool belt. Things like tinderboxes, needles, fishing nets, etc can be added to the tool belt to save space. Just right click an item and if the option "Add to toolbelt" exists, just add it. It saves valuable bank space. If it says that this tool is already on your toolbelt, then dispose of it. What I keep in my bank is simple. My best ranged/mage/melee armor and weapons. A stack of the best food for my level. That's about it.
6. The only way to level Dungeoneering is by raiding floors in Daemonheim. It's pretty simple. First, you obtain the Ring of Kinship and you form a party with it. You can raid floors solo, and I recommend you to do so. Choose a floor which you have unlocked but not yet done. Select the highest complexity possible (always upgrade to the best complexity you can do, maxing out at Complexity 6). Select the size of the floor, either small or medium. I normally do medium, but smalls are fine. How you gain Dungeoneering experience is simple. You can only do so when you successfully complete a floor, which involves defeating a boss for that level.