Learn Polish: To jest mój stary teleskop.

The verb być normally takes the instrumental case when you are defining a subject (like saying On jest moim bratem). However, when you use the pointing word to with jest (meaning 'this is' or 'it is'), the phrase strictly takes the nominative case.

Could I just say To mój stary teleskop without the word jest?

Yes! In everyday Polish, the verb jest is frequently dropped after to when pointing something out or making a simple statement of identity. Both versions are completely natural and correct.

This lesson is about the -ąc participle. Why can't I attach one directly to this sentence?

The -ąc (adverbial contemporary) participle describes a background action done by the subject at the exact same time as the main verb. Because the subject here is the abstract pointing word to (this), it cannot perform an action like 'looking' or 'standing'. You would need a conscious subject, for example: Uśmiechając się, pokazał mi swój stary teleskop (Smiling, he showed me his old telescope).

What gender is teleskop, and how does it affect the words before it?

Teleskop is a masculine inanimate noun. Because the sentence uses the nominative case, both the possessive pronoun mój and the adjective stary must match it by taking their basic masculine nominative singular forms.