I'm making an assumption here, so if I'm incorrect, please let me know and I can delete this answer. I'm assuming that you planted an unnamed variety of watermelon (by "variety", I mean a named cultivar, like Summer Sweet, Yellow Doll, or Sangria). If this is correct, then the small, seedy fruits you're getting from your vines may be due to the parentage of the seeds you planted.
While watermelons cannot cross-pollinate with other cucurbits, they can cross-pollinate with each other. There are several non-sweet/barely edible varieties of watermelon; if one of these crossed with the plant from which your seeds were produced, then your seedlings would have characteristics of both the parents. Here are some possibilities:
- C. lanatus var. citroides (Citron Melon or Bitter Wooly Melon)
- C. caffer. A variety called karkoer has only edible seeds
- C. lanatus mucosospermus (also called C. mucosospermus) is cultivated only for cattle feed
There are other wild varieties that are not considered edible. As you can tell by the list, two of the three varieties are within the C. lanatus species, so seeds labelled "Citrellus lanatus" may contain them without any "false labelling". Also, there are undoubtedly C. lanatus crosses that have undesirable characteristics. Any of these could be a parent of your seeds.
I suggest that you change your seed source, if possible.
To answer your other questions:
1 and 2. Watermelons don't like to be transplanted. If you have a long growing season (100+ days), then I'd recommend just planting them in hills in the soil, about three seeds per hill (about 15cm apart), with the hills at least 3m apart (this depends on variety; some produce shorter vines than others).
Irrigate the plants when the soil is dry at a depth of 8-10cm (about 3 inches).
Watermelons can be heavy feeders, so compost mixed into the soil before planting is a good idea.
There are no suckers.
In my area of the world, watermelons require full sun.