Thank you for your question. Your confusion is understandable. Please take the following as your guide.
SYMBOLS for the chemical elements require a switch to Nemeth Code. Follow Nemeth Code rules regarding use of the English letter indicator. (See Sections 25-30 of the Nemeth Code for the definition of "single letter" and rules regarding use of the English letter indicator.)
Using your examples,
H – Standing alone, a single capitalized letter does require an ELI. See NC Sec.25.a.v and 25.a.vi.
Exception: If H is standing alone within a spatial or tactile molecular structure, the ELI is not used.
(When not standing alone, such as within a chemical formula such as H2O, the ELI is not used.)
(H) – No ELI because the letter is preceded and followed by a sign grouping. See NC Sec.28.a.
Al, Fr, Ag – No ELI because a letter combination that is the same as a shortform does not need an ELI when the first letter is capitalized. See NC Sec.25.b.ii.
Regarding the Periodic table, NC Sec.30 says "When letters appear in tables, whether as entries or headings, the English-letter indicator must be used or must not be used in accordance with the rules contained in §§26-28." In other words, an ELI is needed for single-letter chemical elements in a table.
–Lindy and Kyle
Please do let us know if you have follow-up questions.