Toiletries in hand bag while travelling

The answer will vary, depending upon where you're boarding and where you're going.

In the US, or flying to the US, TSA has this to say about carry-on liquids:

You are allowed to bring a quart-sized bag of liquids, aerosols, gels, creams and pastes in your carry-on bag and through the checkpoint. These are limited to travel-sized containers that are 3.4 ounces (100 milliliters) or less per item. Placing these items in the small bag and separating from your carry-on baggage facilitates the screening process. Pack items that are in containers larger than 3.4 ounces or 100 milliliters in checked baggage.

Any liquid, aerosol, gel, cream or paste that alarms during screening will require additional screening.

As to shaving razors, the same page says:

Disposable Razor

Carry On Bags: Yes

Checked Bags: Yes

Any sharp objects in checked bags should be sheathed or securely wrapped to prevent injury to baggage handlers and inspectors.

I've carried disposable razors (one or two at a time) in my bathroom kit within my carry-on daypack, for years without difficulty or questions.

Note that in all cases, the TSA page warns:

The final decision rests with the TSA officer on whether an item is allowed through the checkpoint.