1. You should run fastqc and see if it indicates that there's adapter contamination. If the fragment sizes were >151 bases (this is quite likely) then you'll have no adapters on the 3' end.
2. Whether the reads overlap or not is irrelevant for most cases. If you want to do assembly then often people like to merge overlapping reads (this helps with error correction). If you're doing differential expression then you just map without any merging because the aligners can handle that without problems. If your original fragments were 100 bases long and your reads are 151 bases, then you're going to want to remove the adapters on each end (or use local alignment and just be done with it).

1) No you cannot assume this. Adapters may or may not be present on each individual read. Normally you would use another program to check the quality of fastq file, and the output tells you which reads include adapters if they are there. I recommend a program called fastqc.

2) I am assuming when you say the "reads overlap" that you mean the read pairs. Please define A) overlapping, B) how you identified how much the reads overlap and C) why overlapping is important to you