Looking for IC which can provide Interrupt signal to detect the Inputs over SPI (3.3V level) (Inputs Expander)

Additionally, my inputs are very fast (not latched - stays ON for about 100ms) so would not like to miss any inputs. It might be possible to have 2 inputs triggered at the same time, is there a way I can accommodate this requirement?

Thank you Clemens Ladisch that really helps.
Is there a way I can reduce some pin usage by using any of the communication master? Ex. I have 2 UART devices, few SPI devices, if I can replace these pins and combine with just one I2C line that would be helpful.

Is there any cost-effective MCU available with 20MHz+ speed which has at least 4 timers on the TI catalog? 
I know Microchip has AVR ATTiny series - 2 for 1$. Is there any cost-effective solution from TI then I would like to use that instead of other options.
https://www.mouser.ca/ProductDetail/Microchip-Technology/ATTINY826-SFR?qs=sGAEpiMZZMv0NwlthflBi775BgPTl04svHdGpOcu4mo%3D

MCU has to be programmable using another MCU over UART/any other protocol.

There are many chips that can be accesses through I²C. Please specify what you want to replace.

This is a search for 20 MHz MCUs with four timers:
www.ti.com/microcontrollers-mcus-processors/microcontrollers/msp430-microcontrollers/products.html#p62=20;25&p2089=4;6

Clemens,

Thank you very much for the help here.

Jigar,

Please let us know what other questions you may have.

Regards,

Eric Hackett 

I would like to use a MCU who I can program using my master MCU. Can you please suggest an option that has that capability right from the root?
I mean, MCU that we can program using another MCU over UART/SPI or any other protocol? And it's been widely proven as working model with examples for it available?

Most of the MSP430s I linked to above can be programmed with UART or I²C; see MSP430 Flash Devices Bootloader (BSL) or MSP430 FRAM Devices Bootloader (BSL).