Many of the concentration spells in 5e revolve around some kind of continuous lockdown, buff, or damaging effect. For example, Bane, Bless, Spirit Guardians, Barkskin, Globe of Invulnerability, etc.

The goals are to prevent spellcasters from being able to apply multiple effects at once and to force players to think carefully about their spell choices, and combat strategy.

For example, in a world without concentration, a cleric could just bless his whole party, bane a bunch of enemies, and run straight into the fray with spirit guardians so that every creature now makes the save with a 1d4 penalty. Heck, he might as well throw a shield of faith on himself too. Wizards could cast hold person, witch bolt the poor paralyzed thing with advantage, and continually zap it over and over again with no fear of it running out of range. Yes, doing these huge combos of spells can cost a lot of spell slots and actions, but it makes smaller, more intimate combat less tactical. Plus, it gives the affected creatures no way to escape their predicament without liberal use of dispel magic and counterspell.

And these problems apply to PCs too. Save or suck spells are only fun when you aren't on the receiving end. Imagine an evil cleric doing what I described above against your party, and it feels really cheap to fight against.

With concentration, you force players to make choices about which continuous effect suits the situation best. It introduces excitement in that these spells can be ended early if the caster fails a concentration save, and it doesn't impede your ability to cast classic one-and-done spells like Magic Missiles, Fire Ball, or Cure Wounds. It also has the benefit of working reasonably well both for and against the PCs.

With that being said, the limited number of higher level spell slots means that concentration isn't as necessary for high level spells. Since a full caster will have six 6th-9th level slots compared to sixteen 1st-5th level slots at 20th level (even fewer high-level spells at lower levels) you can consider dropping the concentration requirement even for continuous effects if you increase the spell level to compensate.

When you introduce custom spells into your campaign setting, I would follow the following guidelines:

1. Pick a published spell that is of a similar power level to what you want to create
2. Note the level and concentration requirements of the published spell and model your spell's requirements around that.
3. If your spell has a continuous effect that may apply every turn, consider adding concentration to its duration if it is below 6th level. If the spell is 6th level or above, feel free to leave out concentration.
4. If you want this spell to synergize with a different concentration spell, make the new spell at least 6th level to compensate.

When it comes to homebrew material, a dose of playtesting will always be required for optimal balance. You should keep an open dialogue with your players about potentially changing the effects of the spells that you add as you see the players using them in game and discover what their impact on the game really is.

No: you do not gain an Ability Score Improvement at each level.

I know there's something called Ability Score Improvement, and I think that means you get to improve 2 ability scores by 1 (or you can add both of them to the same ability) but is this in addition to the 2 you get every level?

There is no "2 ability points that you get every level." That rule does not exist.

Just like any other class feature, an Ability Score Improvement is gained only when it is listed in the Features portion of your class table. With two exceptions, classes in the Player's Handbook gain ASIs at Levels 4, 8, 12, 16, and 19. The Fighter also gains ASIs at Levels 6 and 14; the Rogue gains an additional ASI at Level 10. You can see these listed in each class's experience table.

When you gain an Ability Score Improvement (Basic Rules p. 22 / PHB. p. 59), you may either add +2 to one of your Ability Scores, or you may add +1 to two of your Ability Scores. This may not raise any Ability Score over 20.

As a note: ASIs are gained through individual classes, so if you multiclass into another class instead of continuing to Cleric 4, you will not gain an Ability Score Improvement until you either level up again and choose to advance to your 4th Cleric level or level up enough in your additional class to reach its first ASI at level 4.