Each player has 5 settlement tokens in hand at the beginning of the game. After placing the 5th settlement, considering I have not built a single city, can I build a 6th settlement?
This is super clear in the rules.
Under Build in the Game Rules section:
You cannot build more pieces than what is available in your pool—a maximum of 5 settlements, 4 cities, and 15 roads.
Then, later, under Build (Building) in the Almanac:
You can build as many items and buy as many cards as you desire—as long as you have enough resources to “pay” for them and they are still available in the supply. [...] Each player has a supply of 15 roads, 5 settlements, and 4 cities.
and again later, under Settlements:
Note: If you have built all 5 of your settlements, you must upgrade 1 of your settlements to a city before you can build another settlement. You will then have the settlement in your supply, so you can build another settlement.
Supplies are finite. If you need more settlements, you have to upgrade some to cities. As soon as you upgrade one settlement to a city, the settlement returns to your supply.
The same goes for roads and cities, they too run out. As such, the longest possible road has length 15 (or 30 if you include ships from the Seafarers expansion).
No, unless you have lost a settlement. Check the contents list of the game to see if you have every settlement pawn for that color.