There are a lot of suggestions to do 8-12 reps with short rest for size and less reps with longer rest for strength. Won't my strength increase if I do size workouts with gradual increase of the weights?
Yes. Generally speaking, anything that utilizes your ATP/Creatine Phosphate system will increase both size and strength. The threshold for that system begins at around 70% of your one-rep max.
That said, if strength is your only goal, you would train at rep ranges that purely utilize this system, rather than ranges that also utilize the glycolytic system:
Yes, but it depends on how advanced you are.
If you've been lifting for several years, you'll generally need to focus more and more on one aspect of training in order to see results. If you're just starting out, you'll get stronger/faster/bigger/leaner doing practically any kind of weightlifting. But as those 'newbie gains' taper off, most lifters find they need to program their training to concentrate on specific goals.
Exactly when that transition occurs depends on how often you train, how intensely you train, and your own genetics/hormone levels.
Yeah, but not much.
Training for size will guarantee muscular endurance because you're working on exhausting the muscles in order to grow.
This will increase your strength, but not much.
For example, if you're performing 4 sets of 15 reps of bicep curls with a 30-lb dumbbell, you should be able to perform 2 sets of 5-10 reps with a 35-lb dumbbell. That's an increase in strength.
However, because the focus isn't on strength training, gains in strength will be slower.
Equally important, if strength isn't the goal, your strength will peak. As the weight increases, the number of reps performed will increase. The strength to pump 1 set of 10 reps of bicep curls with a 100-lb dumbbell is much higher than that of a 30-lb dumb bell. Doing 4 sets of those might not be worthwhile for you, especially if your goal can be achieved with smaller weights.
So, when you realize you don't need higher weights to achieve size, you'll stop increasing the weights in order to focus on size.