Yes, but as michaelg said, if it is a small young plant, removing foliage and stems will weaken it. If it is a big strong plant that has been in place several years, remove what you need to remove (within reason).

If you live in a "cold" district, Wisconsin Roses carries it on good multiflora rootstock. If you have mild winters, K&M Nursery carries it on fortuniana rootstock. By the way, another good cut HT is Mavrik. It looks almost like Louise Estes but is just a little different. Lasts for days in a vase . Moonstone and Pop Warner are nothing to turn your back on. Great cut roses. Go to the web site, Roseshow.com. Look under Photo's and look at the HT's. At least half of those 25 HT's make great cut flowers.

My favorite reds are Black Magic and Ingrid Bergman. Black Magic lasts a little longer but Ingrid has a scent. I did get one perfect bloom off Taboo this year but it's been slow to get going and most of the blooms weren't very well formed. I used Moondance as a white cut rose this year. Even though it doesn't last as long as a real cut rose should it bloomed so often that I could afford to cut a new one every few days. And it smells really good too. My favorite cut rose this year is Marilyn Monroe which is more of a cream color. It lasts a long time and has a nice scent but you should be aware it has nasty thorns. I'm ordering a bunch more cut roses from Palatine. They have a really good selection and will state expected vase life. They're roses are really large and healthy too.

Cut the rose's new growth about 2 inches below and above the new growth. Don't use soil, use vermiculite as soil can contain bacteria even sterile soil. Place it is a large water glass that has be rinsed with very hot water to sterilize, and give it a houseplant type of liquid fertilizer like Schultz. Dilute it by half and give a eyedropper ( which is in the bottle) at half the strength.once a month. Place the glass where it will get morning sun only. If it starts to look like it is drying out, put some sort of cover on it like straws and Saran Wrap. This cutting will be very frail as it has started in nice warm conditions. If you want to do a few more, cut all the dead heads off near to the top of the new growth and cut back to new growth. Leave some in the water and even try different methods like the one above. Many ways means you will have more success. The water should be changed if you see roots starting. When you have roots about 3 inches, place it in a container of vermiculite. Be careful when you take the roots out and make a hole with a spoon and gently place the roots in it then cover them up. Roots growing in water are very delicate not like the ones made in soil. They will do fine in vermiculite. Now when you see strong new growth, carefully re-pot in sterile soil. If you want to nitpick, you can sterilize the soil in the oven Here is how. https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/sterilizing-soil.htm