First of all, for a frame shift mutation to occur nucleotides (not amino acids) are either deleted or inserted into the sequence. Read about details on this wiki page. Here's an image about frame shifts.
On panel A a new stop codon is formed (TAG is trasncribed to UAG that is serves as stop codon during translation) thus the result is an abnormally short protein.
On panel B the codon 37 is swapped to a G thus the TA from the codon 38 is shifted left by two resulting in the new codon GTA.
On panel C exactly 3 nuleotides are deleted thus codon boundaries do not change, but a codon and the amino acid encoded by it is completely deleted.
On panel D 4 bases are deleted thus the T form codon 171 and the C from 172 are merged with the G from 170.
source: the linked wiki page.
It does not affect the choromosome in the way you describe because genes are separated and are independently transcribed (with some simplification). Every gene has its own translation start that determines the correct frame in that gene If you think of a chormosome as a chapter of a book and the genes are paragraphs, you can see that deleting letters or words may scrable one paragraph, but the others are not affected by this because they have their own boundaries.
During translation the ribosome reads the mRNA in the form of triplets. When a single nucleotide is deleted then the whole sequence from that point shifts to the left by one thus the first letter of the subsequent triplet gets mixed into the frame where the mutation happened (the same goes for insertion but the sequence is shifted to the right). This causes the change in the meaning of the triplet, and can ruin stop codons that would affect the translation of the mRNA but not the transcription of the gene to mRNA.It is important to understand that transcription is a countinous process and the DNA is transcribed from base to base not in triplets, and that transcriptions stop is not encoded in the from of stop codon.