No, if using a helm of telepathy as it relies on detect thoughts, which doesn't allow this.

You become immune to detection of thoughts at int 3. (PHB 231-2)

If the creature you choose has an Intelligence of 3 or lower or doesn’t speak any language, the creature is unaffected.

As such telepathy doesn't work as you're immune to detect thoughts.

It may work with telepathic bond if you have at least 3 int, such as being a cat.

You forge a Telepathic link among up to eight willing creatures of your choice within range, psychically linking each creature to all the others for the Duration. Creatures with Intelligence scores of 2 or less aren't affected by this spell.

Speak with animals allows communication with less intelligent creatures and says this.

You gain the ability to comprehend and verbally communicate with beasts for the duration. The knowledge and awareness of many beasts is limited by their intelligence, but at minimum, beasts can give you information about nearby locations and monsters, including whatever they can perceive or have perceived within the past day. You might be able to persuade a beast to perform a small favor for you, at the GM’s discretion.

As such, them imposing some penalties on complex communication if you are an animal is reasonable, as a spell that allows you to talk to animals (like horses, spiders, and cats as you said) says you should have limited knowledge and awareness.

From personal experience, it's generally less fun if you require too much thinking about what low intelligence means for players. Some enjoy being dumb some don't and find it emotionally painful. That said, I don't allow complicated communication between players when polymorphed into low int animals because that would lessen the value of druidic wildshape, which explicitly lets your retain mental facilities, and I have found it makes my druid players less happy with their class features.

You can't communicate if using a helm of telepathy or detect thoughts spell as it relies on detect thoughts, which doesn't allow this, but yes for 3 int animals with telepathic bond.

You become immune to detection of thoughts at int 3. (PHB 231-2)

If the creature you choose has an Intelligence of 3 or lower or doesn’t speak any language, the creature is unaffected.

As such telepathy doesn't work as you're immune to detect thoughts.

It may work with telepathic bond if you have at least 3 int, such as being a cat.

You forge a Telepathic link among up to eight willing creatures of your choice within range, psychically linking each creature to all the others for the Duration. Creatures with Intelligence scores of 2 or less aren't affected by this spell.

Speak with animals allows communication with less intelligent creatures and says this.

You gain the ability to comprehend and verbally communicate with beasts for the duration. The knowledge and awareness of many beasts is limited by their intelligence, but at minimum, beasts can give you information about nearby locations and monsters, including whatever they can perceive or have perceived within the past day. You might be able to persuade a beast to perform a small favor for you, at the GM’s discretion.

As such, them imposing some penalties on complex communication if you are an animal is reasonable, as a spell that allows you to talk to animals (like horses, spiders, and cats as you said) says you should have limited knowledge and awareness.

From personal experience, it's generally less fun if you require too much thinking about what low intelligence means for players. Some enjoy being dumb some don't and find it emotionally painful. That said, I don't allow complicated communication between players when polymorphed into low int animals because that would lessen the value of druidic wildshape, which explicitly lets your retain mental facilities, and I have found it makes my druid players less happy with their class features.

No, telepathy relies on detect thoughts, which doesn't allow this.

You become immune to detection of thoughts at int 3. (PHB 231-2)

If the creature you choose has an Intelligence of 3 or lower or doesn’t speak any language, the creature is unaffected.

As such telepathy doesn't work as you're immune to detect thoughts.

Speak with animals allows communication with less intelligent creatures and says this.

You gain the ability to comprehend and verbally communicate with beasts for the duration. The knowledge and awareness of many beasts is limited by their intelligence, but at minimum, beasts can give you information about nearby locations and monsters, including whatever they can perceive or have perceived within the past day. You might be able to persuade a beast to perform a small favor for you, at the GM’s discretion.

As such, them imposing some penalties on complex communication if you are an animal is reasonable, as a spell that allows you to talk to animals (like horses, spiders, and cats as you said) says you should have limited knowledge and awareness.

From personal experience, it's generally less fun if you require too much thinking about what low intelligence means for players. Some enjoy being dumb some don't and find it emotionally painful. That said, I don't allow complicated communication between players when polymorphed because that would lessen the value of druidic wildshape, which explicitly lets your retain mental facilities, and I have found it makes my druid players less happy with their class features.

No, if using a helm of telepathy as it relies on detect thoughts, which doesn't allow this.

You become immune to detection of thoughts at int 3. (PHB 231-2)

If the creature you choose has an Intelligence of 3 or lower or doesn’t speak any language, the creature is unaffected.

As such telepathy doesn't work as you're immune to detect thoughts.

It may work with telepathic bond if you have at least 3 int, such as being a cat.

You forge a Telepathic link among up to eight willing creatures of your choice within range, psychically linking each creature to all the others for the Duration. Creatures with Intelligence scores of 2 or less aren't affected by this spell.

Speak with animals allows communication with less intelligent creatures and says this.

You gain the ability to comprehend and verbally communicate with beasts for the duration. The knowledge and awareness of many beasts is limited by their intelligence, but at minimum, beasts can give you information about nearby locations and monsters, including whatever they can perceive or have perceived within the past day. You might be able to persuade a beast to perform a small favor for you, at the GM’s discretion.

As such, them imposing some penalties on complex communication if you are an animal is reasonable, as a spell that allows you to talk to animals (like horses, spiders, and cats as you said) says you should have limited knowledge and awareness.

From personal experience, it's generally less fun if you require too much thinking about what low intelligence means for players. Some enjoy being dumb some don't and find it emotionally painful. That said, I don't allow complicated communication between players when polymorphed into low int animals because that would lessen the value of druidic wildshape, which explicitly lets your retain mental facilities, and I have found it makes my druid players less happy with their class features.