I think that MSE is much worse than avast.

Your opinion, Ok.

Can MSE protect you on the internet?

Yes. Also, its protection is built-in in IE(SmartScreen Filter).

Can you set-up excludes, create custom scans or schedule boot-time scans?

Yes, but not include boottime scan.

is MSE protect you from known IM infections?

Of course, this is related to virus database, NOT a special shield.

Does It blocks bad websites? What about website reputation?

Again, included in IE(SmartScreen Filter)

How number of viruses have MSE in database?

There are no numbers, only encyclopedia.

Is MSE heuristics good? Isn't It higher than avast?

Yes, see AV-Comparatives tests to stay in tune.

Have It behavioral shield?

Yes. Since MSE 2.0