Mfcttrf

I would like to know how the +1 Keen Composite Longbow (+2 Str) from the dragon's demand works.

I already understand the keen rule, however, I don't know how to apply the bonuses.

My character has a score of 8 in Str (-1), and I understand that this +2 Str bonus plus 2 points to Str score to get 10 gives me a +0, right? Also, does this bonus work for everything (e.g. my abilities), or just when I use the Longbow?

On the other hand, does the +1 get 1 additional to my base attack only, or 1 additional for the damage too? Does this +1 work for all my attacks (e.g. if I get 3 attacks per round)? I'll appreciate if you can give me an example.

A composite longbow +2 Str doesn't give you increased Str, it requires +2 Str. Your 8 Str character will be greatly penalized:

All composite bows are made with a particular Strength rating (that is, each requires a minimum Strength modifier to use with proficiency). If your Strength bonus is less than the strength rating of the composite bow, you can’t effectively use it, so you take a –2 penalty on attacks with it. The default composite longbow requires a Strength modifier of +0 or higher to use with proficiency. A composite longbow can be made with a high strength rating to take advantage of an above-average Strength score; this feature allows you to add your Strength bonus to damage, up to the maximum bonus indicated for the bow. Each point of Strength bonus granted by the bow adds 100 gp to its cost. If you have a penalty for low Strength, apply it to damage rolls when you use a composite longbow.

You will thus suffer a -2 attack penalty when using this bow, and a -1 damage penalty.

However, it's a +1 weapon (+1 to attack and damage with all attacks made using the bow).

Overall, this means you suffer a -1 attack penalty and +0 damage, compared to a regular non-composite longbow. In other words, other than the keen property, this is overall worse for your 8 Str character than a normal longbow.

User Quadratic Wizard has given you a fairly good answer, however i would like to focus on one particular part of your question:

On the other hand, does the +1 get 1 additional to my base attack only or 1 additional for the damage too and does this +1 work for all my attacks? For example do I get 3 attacks per round I'll appreciate if you can give me an example.

You can only have one extra attack per round, unless one of the things granting you an Extra Attack states otherwise, such as the Fighter’s version of it. Page 164 of the Player’s Handbook states:

If you gain the Extra Attack class feature from more
than one class, the features don't add together. You
can't make more than two attacks with this feature
unless it says you do (as the fighter's version of Extra Attack does). Similarly, the warlock's eldritch invocation Thirsting Blade doesn't give you additional attacks if you also have Extra Attack.

Whilst this quote is focusing on multi-classing, the principle is the same for anything else which would grant you an extra attack. You may only have one extra attack unless something states otherwise.