Is it possible to stay overnight in Paris for a flight in London?

This is certainly possible. You will definitely want to book your Eurostar ticket in advance, for two reasons. Firstly, the Eurostar can and does sell out for specific journeys, especially in the summer. Secondly, walk-up tickets for this journey are extremely expensive, around €230. Booking even a fortnight in advance could lower this substantially.

Eurostar has a minimum 30-minute check-in which will include passport control. Allow a little extra time if you're not an EU/EEA passport holder. In this instance, also be aware that the UK is not in the Schengen agreement and visa arrangements differ between France and the UK. You will need to clear UK immigration at Paris Nord.

You'll want a clear two hours at Heathrow plus at least 90 minutes for the transfer from St Pancras to Heathrow (by tube or tube-plus-train). This includes a little safety time. If missing your outbound flight would incur significant extra costs, I'd say allow at least four hours between your planned arrival at London St Pancras and your Heathrow departure.

The direct Piccadilly line tube route to Heathrow does not have air-conditioning and is very warm in July and August. The route via Paddington (Circle/Hammersmith and City line plus TfL Rail stopping train) takes a similar amount of time and is far cooler, but costs a bit more.

Your baggage will be security-cleared at Paris Nord but this is not as arduous as flight regulations - for example, you can carry liquids over 100ml on board so a picnic lunch with drinks for the Eurostar is entirely possible.

A connecting flight may well be quicker and cheaper, despite the inconvenience of getting to a Paris airport. It's worth noting that the journey on Eurostar is not visually spectacular and the tunnel is a bit of a non-event, even boring.