Electronic – How dangerous are disconnected bulging batteries

The limits of "safe" emitted light are very complicated. You can read about the basics here.

A rule of thumb I have heard is if the emitted power is over 5mW, protection should be used. Since the LEDs you linked are capable of 10mW, yes they can be harmful. Do not use them until you understand how they can be harmful and how to prevent it.

UV is particularly dangerous because we can't see it so our blink reflex won't help. To safely work with these LEDs you should get a pair of glasses that block the possible wavelengths the LED can emit, 390 to 405nm ±2.5nm. Examples here.

This paragraph answers an edited out question in the OP wondering why he had a UV pen with no warnings. As for your UV pen light question, it is likely the power was low enough that it was not harmful. Less than 0.39mW (roughly) is considered eye safe so under that no warning would've been required.

Slightly more to-the-point answer concerning the specific materials found in lithium ion batteries:

Lithium metal

Lithium is going to be the number one danger when opening a lithium ion battery. If you get any of it on your skin, the lithium will react with moisture on the skin and ignite more or less on impact, at very high temperature. Counterintuitively, larger amounts of lithium are less dangerous as the hydrogen and other gases produced form a little blanket between the reaction and you. Small specks of lithium can embed themselves on your skin and cause tiny third-degree burns. Lithium dust in your airways can cause havok as well, although the amount needed to really get into trouble is very unlikely to come out of a battery. Only a few types of lithium (ion) batteries contain lithium metal.

Lithium is psychoactive, but you need fairly specific forms of it to be able to absorb this.

Solvents

This is what you smell when dealing with a bad lithium ion battery. The solvents have gotten out. These are actually mostly fine; if you've ever done the brilliantly stupid thing of washing paint stains off your hands with paint thinner, this is about the kind of solvent we're dealing with. They don't combust into particularly bad compounds either, so burning fumes are reasonably safe to deal with.

Fluoride compounds

This is the second big, big red flag. The specific MSDS you supplied has LiPF6 listed, some batteries have even nastier fluorine compounds. These fluorine compounds are generally not stable in air or water and break down to form hydrofluoric acid which is incredibly corrosive. If you breathe in fluorine compounds, they form this acid in your lungs and cause fluid to build up in your lungs, causing shortness of breath. As mentioned in the comments, HF burns are actually painless (as opposed to HCl burns), but still acutely dangerous.

Other compounds

Important to note is that during combustion of the plastics used to make various parts of the battery (sleeve, polymer dielectrics, etc.), there is a chance of small amounts of hydrogen cyanide and chlorine compounds to be emitted. Chlorine compounds form hydrochloric acid in your lungs, whereas cyanide is very potently poisonous. In a well-ventilated room these chemicals aren't particularly harmful, but especially hydrogen cyanide can be a real problem in an unventilated room as it's odorless. Therefore it's very important to always ventilate any combustion reaction involving (long) hydrocarbons.

Medical advice

Never solicit medical advice, or infer any medical advice from EE.SE. If you're at all concerned about what happened, just go see a doctor. He will be able to conclusively tell you if you've suffered any lung (or other) damage from your accident.